Flocks, generalised quadrangles

and translation planes from

BLT-sets

Maska Law

This thesis is presented for the degree of

Doctor of Philosophy of The University of Western Australia Department of Mathematics & Statistics.

March, 2003 ii

Abstract

The study of flocks of the quadratic cone in three dimensional finite projective space is a fascinatingly rich field within finite geometry. The diversity of the geometric structures intimately connected with flocks lends great appeal, and these connections play a major role in this thesis. In particular, we study the case of flocks over fields of odd characteristic, where we have available the construction of Bader, Lunardon and Thas of a set of flocks from a given flock, and the concept of a BLT-set.

Following a survey of finite projective spaces and finite polar spaces, we study translation planes arising from linespreads of three dimensional projective space via the constructions of André and of Bruck and Bose, and we study elation generalised quadrangles via the group coset geometry construction of Kantor. The construction of Thas and Walker of an ovoid of the Klein quadric from a flock of the quadratic cone connected the extensive theory of translation planes with the study of flocks. The introduction by Payne of the concept of a q-clan for use in the construction of Kantor provided the opportunity for Thas to describe the connection between flocks and elation generalised quadrangles. We survey these many connections, introducing the known infinite families as examples of the various structures. We also apply the so called Fundamental Theorem of q-clan geometry to the calculation of the automorphism group of an example.

The concept of a BLT-set, introduced by Bader, Lunardon and Thas, but named by Kantor, provided a more general model in which to study flocks over fields of odd order, since one BLT-set gives rise to a number of flocks, which may or may not be pairwise equivalent. By applying results of Bader, O’Keefe and Penttila, we are able to exploit some recent upgrades to the computer algebra system MAGMA in order to search for new examples of BLT-sets. We discuss the general ideas behind these searches, and present a total of 31 new BLT-sets of varying order, yielding 170 new flocks. Analysis of one of the new examples led to the construction of an infinite family of BLT-sets over fields of characteristic three. These have been shown by Payne to be new, and give rise to the first known infinite family of flocks with trivial automorphism group. As well as each of the known infinite families of BLT-sets, we present a complete list of all known BLT-sets not currently belonging to an infinite family, and hence cover all BLT-sets known at this time. iv

Contents

Abstract iii

Preface vii

Acknowledgements ix

1 Preliminaries and background 1 1.1 Finite projective spaces and finite polar spaces . . . . . . . . . . . . . 1 1.2 Orthogonal spaces and quadrics . . . . . . . . . . . . . . . . . . . . . 16

1.2.1 The Klein correspondence . . . . . . . . . . . . . . . . . . . . 22

2 Translation planes and elation generalised quadrangles 25

2.1 Projective planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1 Translation planes . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2 Generalised quadrangles . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.1 The classical generalised quadrangles . . . . . . . . . . . . . . 33 2.2.2 The constructions of Tits . . . . . . . . . . . . . . . . . . . . . 33 2.2.3 Elation generalised quadrangles . . . . . . . . . . . . . . . . . 35

3 Flocks of circle planes 39 3.1 Flocks of quadratic cones . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Associated geometries 45

4.1 Translation planes arising from linespreads . . . . . . . . . . . . . . . 45

4.1.1 Flocks and translation planes . . . . . . . . . . . . . . . . . . 49 4.2 Elation generalised quadrangles arising from q-clans . . . . . . . . . . 51

4.2.1 Flocks, translation planes and elation generalised quadrangles 54 4.3 Fundamental Theorem of q-clan geometry . . . . . . . . . . . . . . . 55

4.4 Hyperbolic fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Application of the Fundamental Theorem of q-clan geometry 61 5.1 A semifield example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 BLT-sets 69 6.1 Derivation of flocks and BLT-sets . . . . . . . . . . . . . . . . . . . . 70 6.2 BLT-sets, generalised quadrangles and translation planes . . . . . . . 74 6.3 The known infinite families of BLT-sets . . . . . . . . . . . . . . . . . 79

6.3.1 An alternative model of Q(4, q) . . . . . . . . . . . . . . . . . 85

7 Searching for BLT-sets 91 7.1 Search techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8 The new BLT-sets 99 8.1 The new examples for q ≤ 89 . . . . . . . . . . . . . . . . . . . . . . 101 8.2 The new example for q = 125 . . . . . . . . . . . . . . . . . . . . . . 118

9 Analysis of some examples 121 9.1 An infinite family of BLT-sets in characteristic 3 . . . . . . . . . . . . 121 9.2 Possibly sporadic examples . . . . . . . . . . . . . . . . . . . . . . . . 123

Bibliography 125

Index 135

Preface

The concept of a BLT-set arose in the field of finite geometry in 1990 with the introduction by Bader, Lunardon and Thas of the process of derivation of flocks of the quadratic cone of PG(3, q). The study of such flocks had constituted a very active area of research for many years before this, due in large part to the connections with other extensive fields of research, namely, the studies of certain translation planes and elation generalised quadrangles. These connections are a major motivation behind the continued interest in flocks of the quadratic cone, and so the focus on BLT-sets in this thesis requires the wider context of flocks and their associated geometries. An attempt is made to cover some important parts of this wide setting, but no claims to completeness are made in any respect. The contents of Chapters 1-4 and 6 make up this wider context, and full references are given for all ideas covered therein. Perhaps, however, some topics are presented in a fashion other than that which has become standard in the literature; for example, most of the known infinite families are presented in the setting in which they were originally described, and then compiled into a list after the wider context has been covered. The exceptions are presented in the models which the author considers most suitable for their presentation, but this is, of course, a matter of taste.

The contribution to current knowledge claimed by this thesis forms the content of Chapters 7-9, where results and outcomes of computer searches for BLT-sets are presented. The ideas underlying these searches, which are discussed in Chapter 7, are those of the author’s supervisor Tim Penttila. The successful searches, undertaken by the author and their supervisor during the years of the author’s candidature, yielded a wealth of new examples of BLT-sets, and these are presented in Chapter 8. Analysis of one of these examples led to the construction of a new infinite family of BLT-sets, which is presented in Chapter 9. Along with the new examples of BLT-sets, all previously known infinite families and all previously known examples which do not yet belong to an infinite family are presented in this thesis, and in so doing, all presently known BLT-sets are covered (to the author’s best knowledge).

The initial content of Chapter 5 is in some sense well known material also, however the calculation given here has not appeared in the literature. An equivalent calculation in a different model has been given elsewhere, but the theme of this chapter is to demonstrate an understanding of the difficult topic it covers, and as for the presentation of the wider context earlier, to present in detail an important topic related to flocks and their associated geometries. viii

Acknowledgements

I’d like to say a personal thankyou to those people who have made my time as a research student a very enjoyable and rewarding experience.

Thankyou to my supervisor Tim Penttila, for introducing me to the beautiful world of finite geometry, and for teaching me so much, albeit so little of your vast knowledge. Thankyou for all the opportunities you have provided me, and just for being a great supervisor. It has been a real pleasure talking and learning and hanging out with you over the years.

Thankyou to Christine O’Keefe for support at local and international conferences, and for the Adelaide workshops. To Stan Payne and Bill Cherowitzo in Denver, to Guglielmo Lunardon and Laura Bader in Napoli, and to Jef Thas and Frank De Clerck in Gent, thankyou all for your warm hospitality during my visits to your universities. It is a great encouragement to know there are so many nice people in this area. Particular thanks to Stan Payne for the lectures you gave whilst I was in Denver and for your enthusiastic interest in my work, and to Guglielmo Lunardon, Laura Bader, Nicola Durante and Tim Penttila for showing me how much fun collaboration can be.

Thankyou to my good friend Michel Lavrauw, for your encouraging words on mathematics, and life in general, and for enjoying a good laugh. To Nicholas Hamilton, Matt Brown, Leanne Holder, Nico Durante and Rocco Trombetti for some fun times in the name of mathematics and otherwise.

Thankyou to everyone in the Maths and Stats Department for being part of a productive, and sociable, working environment.

Thankyou to my wonderful family, Karla, Malcolm and Saskia, for always being there. Thankyou for our great times together, and for encouraging my education over the years. Thankyou Judit for always believing in me, and for your immense support during the difficult times. To all my fabulous friends, cheers for being the great people you are.

And finally, thanks to Barrett’s Bread for making the best coffee around ! x

CHAPTER 1 Preliminaries and background

In this thesis we will be studying finite geometries. Some good sources of reference for preliminaries and background include Artin (1957), Baer (1952), Beukenhout (Ed.) (1995), Cameron (year unknown), Dembowski (1968), Hirschfeld (1979, 1985, 1998), Hirschfeld and Thas (1991) and Taylor (1992).

1.1 Finite projective spaces and finite polar spaces

We will be concerned with finite dimensional projective geometries over finite fields. For a positive integer n and a prime power q, the subspaces of the n-dimensional projective space PG(n, q) are the subspaces of a vector space V of rank rk = n + 1 over the Galois field GF (q) of order q. We view PG(n, q) as an incidence structure with symmetric inclusion as the incidence relation. So that we retain familiar geometric properties, the dimension of an element of a projective space is defined to be one less than its rank as a vector subspace. Thus the points of a projective space are the vector subspaces of rank 1, hence of dimension 0. Subspaces of codimension 1 are called hyperplanes.

Building on work of von Staudt and of Hilbert, Veblen and Young (1916) gave axioms for a projective geometry solely in terms of the points and lines of the space : for a set of points and a set of distinguished subsets of the set of points called lines, suppose

(i) any two points lie on a unique line;

(ii) any line contains at least three points;

(iii) any line which meets two sides of a triangle, not at a vertex, meets the third side also.

A subspace is a subset S of points such that for any two points P, Q in S, all points on the unique line P Q are also in S. The projective geometry is then the incidence structure of all nonempty subspaces satisfying these axioms, with incidence inclusion. A projective geometry has dimension d if there exists a chain of proper nonempty subspaces of length d, but not of length d + 1.

In fact, any finite projective geometry of dimension at least 3 is isomorphic to PG(n, q) for some n and q. But for dimension 2, there exist numerous projective planes not isomorphic to PG(2, q), satisfying the first two axioms and the simpler condition that any two lines meet. The planes PG(2, q) are called Desarguesian as they are the only finite projective planes which satisfy the theorem of Desargues. For n ≥ 3, PG(n, q) is always Desarguesian.

It follows from the axioms that any two lines of a projective geometry have the same number of points. Since, consider two lines l, l′ which meet in some point P. Take a third line m meeting both l and l′ in points other than P. By axiom (ii), there is a third point Q on m. Any line through Q which meets l in a point R will also meet l′ in some point R′, by axiom (iii). In this way, the points of l other than P are uniquely paired with the points of l′ other than P. If on the other hand the lines l and l′ do not meet in a point, take a point P on l and a point P′ on l′. Then the unique line P P′ has the same number of points as l by the above argument, but also has the same number of points as l′, and so the result follows. If the number of points on a line of the projective geometry PG is s + 1, we call s the order of PG. The order of PG(n, q) is q.

As with many mathematical objects, we are interested in automorphisms of projective geometries, that is, structure-preserving bijections of the space. A collineation of a projective geometry is a permutation of the points which maps lines onto lines, and thus maps subspaces onto subspaces. So a collineation is an inclusion-preserving bijection of the space. The set of all collineations of a projective geometry PG forms a group under composition called the collineation group (or automorphism group) and is denoted by AutPG.

A theme that will become very important in the following sections is that of transferring information between a projective space and the underlying vector space (in the Desarguesian case), and we begin this theme in the context of collineations.

For a vector space V over GF (q), a transformation T : V → V is called αsemilinear if there exists a field automorphism α of GF (q) such that

T (v + w) = T (v) + T (w) and T (cv) = α(c)T (v)

for all v, w ∈ V and c ∈ GF (q). We call α the companion automorphism of T. Clearly, any semilinear transformation of V will map subspaces onto subspaces, and thus any invertible semilinear transformation will give rise to (or induce) a collineation of the associated projective space PG(n, q). The set of all invertible semilinear transformations forms a group under composition and is denoted by Γ L(n + 1, q).

Now the elements of Γ L(n + 1, q) which induce collineations fixing every point of PG(n, q) form the subgroup

Z = {v ↦ cv | c ∈ GF (q), c ≠ 0},

that is, the pointwise stabiliser of PG(n, q) is the set of collineations induced by the scalar maps. Thus the group of collineations of PG(n, q) induced by the invertible semilinear transformations is P Γ L(n+1, q) = Γ L(n+1, q)/Z, called the projective semilinear group.

Conversely, if the rank of V is at least 3, then it can be shown that every collineation of PG(n, q) has this form, that is, every collineation is induced by an invertible semilinear transformation of V. Thus we have the following result.

Theorem 1.1 (The Fundamental Theorem of Projective Geometry). If the rank n + 1 of V is at least 3, then AutPG(n, q) = P Γ L(n + 1, q).

Proof : See Baer (1952), p44.

A very interesting feature of a projective geometry PG is that it satisfies a duality concept. A proposition concerning PG is often stated only in terms of subspaces and inclusion relations between them, for example, meet, join or dimension. The dual proposition is then obtained by inverting all inclusions, and hence replacing meet, join and dimension r with join, meet and dimension n − 1 − r, respectively (if PG has dimension n). This concept is referred to as the Principle of Duality : for a proposition which is true in all n-dimensional projective geometries, the dual proposition is also true in all n-dimensional projective geometries.

Motivated by the Principle of Duality, we introduce the concept of a duality (or correlation) of a projective geometry, that is, an inclusion-reversing bijection of the space. So a duality δ of a projective geometry PG is a permutation of the subspaces such that S ⊆ T ⇒ S^{δ} ⊇ T^{δ}, for all subspaces S and T of PG. The product of two dualities is then a collineation, and the set of all dualities and collineations forms a group under composition, called the correlation group and denoted by CorPG.

The most important type of duality is one of order 2, called a polarity of the projective geometry. The reason for their importance is that they give rise to particular geometries called polar spaces, which are similar in nature to projective geometries, yet rich in different types of structure.

Let ρ be a polarity of a Desarguesian projective space PG(n, q). A subspace S is totally isotropic with respect to ρ if S ⊆ S^{ρ}. The polar space Π(ρ) arising from ρ is the geometry whose subspaces are the totally isotropic subspaces of PG(n, q) with respect to ρ, with incidence inherited from PG(n, q). We call S^{ρ} the polar subspace of the subspace S with respect to ρ.

We return to our theme motivated by the Fundamental Theorem of Projective Geometry, and ask what information we can obtain by studying the transformations on the underlying vector space which give rise to polarities.

A sesquilinear form on a vector space V is a function f : V × V → GF (q) which is linear in the first variable and semilinear in the second variable, that is, for which there exists a field automorphism α such that

f(v + v′, w) = f(v, w) + f(v′, w) f(cv, w) = cf(v, w)
f(v, w + w′) = f(v, w) + f(v, w′) f(v, cw) = α(c)f(v, w),

for all v, v′, w, w′ ∈ V and c ∈ GF (q). We call α the companion automorphism of f, and say f is an α-sesquilinear form. If α is the identity, then the form is called bilinear.

The radical of a sesquilinear form f is

Rad(f) = {w ∈ V | f(v, w) = 0 for all v ∈ V }.

A sesquilinear form is nondegenerate if its radical is {0}.

Consider the vector space V = GF (q)^{n}. If A is an n × n matrix with entries from GF (q) and α ∈ AutGF (q), then f(v, w) = v^{T} Aα(w) is an α-sesquilinear form on V. Conversely, if f is an α-sesquilinear form on V = GF (q)^{n}, X = {x_{1}, …, x_{n}} is a basis for V, and A = A(X, f) is the n × n matrix A = (a_{ij}) with a_{ij} = f(x_{i}, x_{j}), then the isomorphism GF (q)^{n} → V taking (a_{1}, …, a_{n}) to a_{1}x_{1} +⋯+a_{n}x_{n} identifies (v, w) ↦ v^{T} Aα(w) with f. The matrix A = A(X, f) is called the matrix of f with respect to the basis X. Note that A (and α) uniquely determine the form f.

Lemma 1.1. A sesquilinear form f on a vector space V is nondegenerate if and only if the matrix of f with respect to a basis of V is nonsingular.

Proof : If A is the matrix of f with respect to some basis of V and α is the companion automorphism of f, then the radical of f corresponds to the kernel of the map x ↦ Aα(x). Since this map is a bijection, the kernel is {0} if and only if A is nonsingular.

We say two vectors v, w are orthogonal (with respect to f) if f(v, w) = 0. For a subset X of V, we define the orthogonal complement of X (with respect to f) to be

X^{⊥} = {v ∈ V | f(x, v) = 0 for all x ∈ X},

where the symbol ⊥ is read as ‘perp’. So a subspace U of V is nondegenerate if the restriction of f to U is nondegenerate, or equivalently, if U ∩ U^{⊥} = {0}.

For a nondegenerate sesquilinear form f on a vector space V, and U a subspace of V, other familiar properties also hold :

(i) rk U + rk U^{⊥} = rk V; (ii) U is nondegenerate if and only if V = U ⊕ U^{⊥}; (iii) (U^{⊥})^{⊥} = U.

Thus the map PG(n, q) → PG(n, q) with U ↦ U^{⊥}, for all subspaces U of V, is a duality of the projective space PG(n, q). We call such a duality the duality arising from f. It is determined up to a nonzero scalar multiple of f.

Theorem 1.2 (Birkhoff and von Neumann (1936)). Every duality of PG(n, q), n ≥ 2, arises from a nondegenerate sesquilinear form on the underlying vector space, up to a nonzero scalar multiple.

Proof : See Baer (1952), p102.

The previous theorem was first proved by Birkhoff and von Neumann in 1936 whilst attempting to provide a logical structure for such physical theories as quantum mechanics. As a consequence of the theorem we see that the study of dualities of a projective space is equivalent, up to a nonzero scalar multiple, to the study of nondegenerate sesquilinear forms on the underlying vector space.

A sesquilinear form f on V is reflexive if the orthogonality relation is symmetric, that is, if for all v, w ∈ V,

f(v, w) = 0 ⇒ f(w, v) = 0.

Lemma 1.2. The duality arising from a nondegenerate sesquilinear form f is a polarity if and only if f is reflexive.

Proof : Let δ be the duality arising from f, so that

δ(⟨v⟩) = v^{⊥} = {w | f(v, w) = 0}.

Then

δ is a polarity ⟺ δ^{2} = 1
⟺ (⟨v⟩ ⊆ δ(⟨w⟩) ≡ ⟨w⟩ ⊆ δ(⟨v⟩)) ∀v, w ∈ V
⟺ (f(w, v) = 0 ≡ f(v, w) = 0) ∀v, w ∈ V
⟺ f is reflexive.

So in order to study polarities of a projective space, we need only concern ourselves with nondegenerate reflexive sesquilinear forms defined on the underlying vector space.

Before continuing our study of sesquilinear forms, we consider the question of isomorphism of vector spaces carrying such forms.

So let V_{1}, V_{2} be vector spaces of the same dimension carrying the sesquilinear forms f_{1}, f_{2}, respectively. Let T : V_{1} → V_{2} be an invertible semilinear transformation with companion automorphism α. Then T is called a semisimilarity if there exists c ∈ GF (q) \ {0} with

f_{2}(T (v), T (w)) = cα(f_{1}(v, w))

for all v, w ∈ V_{1}. If α is the identity, then T is linear and is called a similarity. A similarity for which the constant of similitude c is the multiplicative identity of GF (q) is called an isometry. We say two spaces are isometric (similar) if there exists an isometry (similarity) between them. Note that an isometry preserves the sesquilinear forms on the spaces, whereas a similarity preserves only orthogonality of vectors.

Lemma 1.3. A sesquilinear form g on V is similar to f if and only if there exists a basis Y = {y_{1}, …, y_{n}} of V with A(Y, g) = cA(X, f) for some c ∈ GF (q) \ {0}. The forms are isometric precisely when c can be chosen to be 1.

Proof : See Aschbacher (1986), p78.

The set of all semisimilarities from a vector space V with form X to itself forms a group under composition, called the semisimilarity group Γ X(V) of V. It is clear from the definitions that the map from a semisimilarity to its companion automorphism is a homomorphism onto AutGF (q) with kernel the similarity group GX(V). Likewise, the map from a similarity to its constant of similitude c is a homomorphism onto GF (q) \ {0} with kernel the isometry group IX(V) (or simply X(V)).

An α-sesquilinear form f on a vector space V is called α-hermitian if f(v, w) = α(f(w, v)) for all v, w ∈ V. α has order at most 2. If α is the identity, then f (which is bilinear) is called symmetric. A bilinear form f on V is called alternating if f(v, v) = 0 for all v ∈ V.

Theorem 1.3 (Classification of nondegenerate reflexive sesquilinear forms). A nondegenerate reflexive sesquilinear form is either alternating, symmetric or similar to an α-hermitian form with α ≠ 1.

Proof : See Baer (1952), or Cameron (year unknown).

Traditionally, a vector space carrying an alternating form is called symplectic, and the polarity arising from this space is a symplectic (or null) polarity. However, a short calculation (expand f(v + w, v + w) = 0) shows that an alternating bilinear form over a field of characteristic 2 is also symmetric, whereas a symmetric bilinear form need not be alternating. In order to distinguish these cases, over a field of characteristic 2, we call a polarity symplectic (or null) if it arises from an alternating symmetric bilinear form (as before), and pseudo-symplectic if it arises from a non-alternating symmetric bilinear form. For characteristic not 2, a vector space carrying a symmetric bilinear form is called orthogonal and gives rise to an orthogonal polarity. A vector space over an arbitrary field carrying a hermitian sesquilinear form is called unitary and gives rise to a unitary polarity. These names correspond to the names of the classical isometry groups associated with such spaces.

As we now know that every polarity arises from a nondegenerate reflexive sesquilinear form, we would like to construct the polar space arising from a polarity directly from the underlying vector space and form.

For a vector space V with nondegenerate sesquilinear form f we say a nonzero vector v ∈ V is isotropic if f(v, v) = 0. A subspace U is totally isotropic if the restriction of f to U is trivial, that is, if any two vectors are orthogonal (f(v, w) = 0 for all v, w ∈ U). This is equivalent to saying that U ⊆ U^{⊥}, and so we see that this definition of totally isotropic is consistent with the earlier usage. A subspace is anisotropic if it contains no nonzero isotropic vectors. A subspace of the projective space arising from V is called totally isotropic if it is totally isotropic as a subspace of V.

The polar space arising from a vector space with nondegenerate reflexive sesquilinear form is the geometry whose subspaces are the totally isotropic subspaces of the associated projective space, with incidence inherited from the projective space.

Now this is not the only way in which these sorts of geometries arise. A geometry with properties akin to the above polar spaces can be constructed from a nondegenerate quadric of projective space, and as we shall see, is indeed one of these geometries in odd characteristic, but gives rise to new examples in characteristic 2.

A quadratic form on a vector space V is a function Q : V → GF (q) which satisfies

Q(cv) = c^{2}Q(v)

for all v ∈ V and all c ∈ GF (q), and has associated with it a symmetric bilinear form, f_{Q}, where

f_{Q}(v, w) = Q(v + w) − Q(v) − Q(w)

for all v, w ∈ V. We say that f_{Q} is the polar form of Q or that Q polarises to f_{Q}.

The singular radical of a quadratic form Q is

SingRad(Q) = {v ∈ Rad(f_{Q}) | Q(v) = 0}.

A quadratic form is nondegenerate if its singular radical is {0}.

We note that the question of isomorphism of vector spaces with quadratic forms is analogous with that of spaces carrying sesquilinear forms, and results carry over naturally from the earlier discussion.

A quadric Q of PG(n, q) arising from the quadratic form Q is the set of points ⟨v⟩ such that Q(v) = 0. We say a quadric is nondegenerate if the defining quadratic form is nondegenerate.

The polar space Π(Q) arising from a nondegenerate quadric Q is the geometry whose subspaces are the subspaces of PG(n, q) contained in Q, with incidence inherited from PG(n, q).

Again, we can construct the polar space arising from a quadric directly from the underlying vector space and quadratic form.

For a vector space V with nondegenerate quadratic form Q, we say a nonzero vector v ∈ V is singular if Q(v) = 0. A subspace U is totally singular if the restriction of Q to U is trivial, that is, if Q(v) = 0 for all v ∈ U. If a subspace is totally singular, then it is clear that the subspace is also totally isotropic with respect to the polar form of the quadratic form, since the quadratic form uniquely determines its polar form. A subspace is anisotropic if it contains no nonzero singular vectors.

A subspace of the projective space arising from V is called totally singular if it is totally singular as a subspace of V.

The polar space Π(Q) arising from a nondegenerate quadratic form Q on a vector space is the geometry whose subspaces are the totally singular subspaces of the associated projective space.

It was mentioned above that, under certain circumstances, this type of polar space actually coincides with the case of a polar space arising from a polarity. For the polar form f_{Q} of a quadratic form Q, we have that f_{Q}(v, v) = 2Q(v) for v ∈ V. So if the characteristic of GF (q) is not 2, the quadratic form Q is uniquely determined by f_{Q}, and so the study of quadratic forms is equivalent to the study of symmetric bilinear forms. However, if the characteristic of GF (q) is 2, the polar form of a quadratic form is alternating, and so for quadrics in characteristic 2 we get the polar spaces arising from symplectic polarities, but not those arising from pseudosymplectic polarities. Furthermore, we cannot recover the quadratic form from the polar form. Indeed, there are many quadratic forms which could give rise to a given alternating symmetric bilinear form as its polar form.

For this reason, we intend to no longer consider vector spaces carrying symmetric bilinear forms, except when they arise as the polar form of a quadratic form. Thus for characteristic 2 we intend to exclude the possibility of a pseudo-symplectic polarity. However, it turns out that the polar space arising from a pseudo-symplectic polarity is isomorphic to the polar space arising from a symplectic polarity in dimension one less. Hence in general we will only consider the following cases :

(i) a nondegenerate alternating bilinear form (giving rise to a symplectic polarity)

(ii) a nondegenerate α-hermitian sesquilinear form with α ≠ 1 (giving rise to a unitary polarity)

(iii) a nondegenerate quadratic form with symmetric bilinear polar form (giving rise to an orthogonal polarity for characteristic not 2, and a symplectic polarity for characteristic 2).

For uniformity, a vector space carrying a form of one of these types will be called a space with form, of type symplectic, unitary or orthogonal, respectively, and the families of polar spaces arising in these situations are called the classical polar spaces.

Now we investigate more closely the structure of the finite classical polar spaces, by studying the underlying spaces with form.

The following is a powerful result from the theory of categories, first proved by Witt in 1937 for symmetric and hermitian forms over fields of characteristic not 2.

Theorem 1.4 (Witt (1937)). Suppose V is a space with form, U is a subspace of V and the map T : U → V is an isometry. Then T extends to an isometry of V.

Proof : See Aschbacher (1986), p81.

As a consequence of Witt’s theorem, we can define an invariant of spaces with form. Consider two maximal totally isotropic subspaces M and N of a vector space V. Without loss of generality, suppose that rk M ≤ rk N. Then there is an isomorphism S of M into N, and this isomorphism is in fact an isometry since M and N are totally isotropic. So by Witt’s theorem, there is an extension of S to an isometry T of V onto V. Then M ⊆ T^{−1}(N); but T^{−1}(N) is totally isotropic. Thus M = T^{−1}(N), and so rk M = rk N. Hence all maximal totally isotropic subspaces have the same rank. This constant rank is called the Witt index of the space, and since a totally isotropic subspace is one which is contained in its perp, the Witt index can be at most half the rank of the underlying vector space. A similar argument holds for maximal totally singular subspaces, and so in this case, the Witt index will be equal to one more than the dimension of a maximal subspace contained in the quadric.

With this invariant of a space with form we are able to completely describe the structure of these spaces, and hence describe all classical polar spaces of this type. Firstly, we require some more geometric concepts.

A pair of vectors (u, v) in V such that u and v are isotropic, and Q(u) = Q(v) = 0 in the orthogonal case, is called a hyperbolic pair if f(u, v) = 1. The line ⟨u, v⟩ in PG(n, q) is then called a hyperbolic line.

If V = U ⊕ W and f(u, w) = 0 for all u ∈ U and w ∈ W, we write V = U ⊥ W and say that V is the orthogonal direct sum of U and W.

Lemma 1.4. Suppose that l is a nondegenerate subspace of V of rank 2 which contains an isotropic vector u with respect to the sesquilinear form f. Then l = ⟨u, v⟩, where (u, v) is a hyperbolic pair.

Proof : See Taylor (1992), p56.

Lemma 1.5. (i) Any two symplectic hyperbolic lines over the same field are isometric. (ii) Any two unitary hyperbolic lines over the same field are isometric. (iii) Any two orthogonal hyperbolic lines over the same field are isometric.

Proof : The matrix of an alternating or hermitian form f with respect to the basis {u, v} of the hyperbolic line l = ⟨u, v⟩ is

[ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ]

For a quadratic form Q, we have Q(au + bv) = 0. Since these remarks hold for any form and any hyperbolic line, the result follows from Lemma 1.3 and the analogous result for spaces with quadratic forms.

Lemma 1.6. For vector spaces U_{1}, U_{2}, W_{1}, W_{2}, if U_{1} is isometric to U_{2} and W_{1} is isometric to W_{2}, then the orthogonal direct sum U_{1} ⊥ W_{1} is isometric to the orthogonal direct sum U_{2} ⊥ W_{2}.

Proof : This follows immediately from the definition of orthogonal direct sum.

The following result describes completely the structure of a vector space with form in terms of the previous concepts.

Theorem 1.5. Let V be a space with form. Then V is an orthogonal direct sum of r hyperbolic lines and an anisotropic space U, where r is the Witt index of V. U is determined up to isometry.

Proof : (Existence) We show existence of an orthogonal direct sum of the required form by induction on the Witt index r. If r = 0, then V has no maximal totally isotropic subspaces other than {0}, so V is anisotropic by definition. So suppose that r > 0. Then since there is a maximal totally isotropic subspace of rank r > 0, totally isotropic points must exist. Consider such a point P. Then P^{⊥} is a hyperplane, and so there exists a line l on P with l ⊈ P^{⊥}. Let R = l ∩l^{⊥} and take x ∈ R. Then since x ∈ l^{⊥}, l ⊆ x^{⊥}, and so P ⊆ x^{⊥}. Thus x ∈ P^{⊥}. Now l ⊈ P^{⊥}, so l ∩ P^{⊥} = P. So we have x ∈ P. But if x ≠ 0, then x^{⊥} = P^{⊥}; and l ⊆ x^{⊥}, so l ⊆ P^{⊥}. This contradicts the original assumption, so x = 0. Thus R = {0}, that is, l is nondegenerate. So nondegenerate lines on totally isotropic points exist. By Lemma 1.4, all such lines are hyperbolic, and so hyperbolic lines exist. Now consider l^{⊥} for a hyperbolic line l on a totally isotropic point P. Since l^{⊥} ∩ (l^{⊥})^{⊥} = l^{⊥} ∩ l = {0}, we see that l^{⊥} is nondegenerate.

Our next step is to show that the Witt index of l^{⊥} is r − 1 : Firstly, since P ⊈ l^{⊥}, the span of P and any totally isotropic subspace of l^{⊥} will give a totally isotropic subspace of V with rank at most r. So the Witt index of l^{⊥} is not greater than r − 1. Secondly, let M be a maximal totally isotropic subspace of V on P. Take w ∈ l such that w ∉ P. Then M ∩ l^{⊥} ⊆ M ∩ w^{⊥}. Now suppose P = ⟨v⟩ and so l = ⟨v, w⟩. Consider u ∈ M ∩ w^{⊥}. Then since M is totally isotropic, M ⊆ M^{⊥}; and since P ⊆ M, we have that M ⊆ P^{⊥}. So f(u, v) = 0 for the defining form f. Also, since u ∈ w^{⊥}, f(u, w) = 0. So we have that f(u, λv + μw) = 0, and so u ∈ l^{⊥}. Thus M ∩ w^{⊥} ⊆ M ∩ l^{⊥}. Hence M ∩ l^{⊥} = M ∩ w^{⊥}. Now w^{⊥} is a hyperplane, and if M ⊆ w^{⊥} then f(v, w) = 0; but f(v, w) ≠ 0 since l = ⟨v, w⟩ is a hyperbolic line, and so M ⊈ w^{⊥}. Hence the rank of M is one more than the rank of M ∩ w^{⊥}, that is, rkM = r = 1 + rk(M ∩ w^{⊥}). Thus rk(M ∩ l^{⊥}) = r − 1. So a maximal totally isotropic subspace of l^{⊥} must have rank not less than r − 1. Thus the Witt index of l^{⊥} is r − 1.

Now since l is nondegenerate, V = l ⊕ l^{⊥}. Our inductive hypothesis says that the result holds for a nondegenerate space of Witt index r − 1; in particular, for l^{⊥}. That is, l^{⊥} is an orthogonal direct sum of r − 1 hyperbolic lines and an anisotropic subspace. Hence V = l ⊕ l^{⊥} will be an orthogonal direct sum of r hyperbolic lines (since l is hyperbolic) and an anisotropic subspace.

(Uniqueness) By Lemmas 1.5 and 1.6 any two orthogonal direct sums of r hyperbolic lines of V are isometric. By Witt’s theorem, there is an isometry T of V taking the first U_{1} of these sums to the second U_{2}. So T will take (U_{1})^{⊥} to (U_{2})^{⊥}. Since each of these sums is nondegenerate, we have that V = U_{1} ⊥ (U_{1})^{⊥} and V = U_{2} ⊥ (U_{2})^{⊥}. So (U_{1})^{⊥} and (U_{2})^{⊥} are in fact the anisotropic spaces related to each of the sums. Thus any two anisotropic spaces are isometric.

Note that the hyperbolic lines play a similar role here to an orthonormal basis in a real inner product space. If a space is just an orthogonal direct sum of hyperbolic lines, that is, the anisotropic part is just the zero subspace, then we call it a hyperbolic space. Our final step to describing completely the structure of our spaces with form is to determine all possible anisotropic parts of a given space.

Lemma 1.7. An anisotropic symplectic space has rank 0.

Proof : All nonzero vectors are isotropic under an alternating form.

Corollary 1.1. A symplectic space is isometric to an orthogonal direct sum of r hyperbolic lines, and thus has even rank 2r, where r is the Witt index. This space is denoted by Sp(2r, q).

Lemma 1.8. An anisotropic unitary space has rank at most 1, and all anisotropic unitary spaces of rank 1 are isometric.

Proof : See Taylor (1992), p116, or Cameron (year unknown), pp76-77.

Corollary 1.2. A unitary space of even rank 2r over the field GF (q^{2}) is isometric to an orthogonal direct sum of r hyperbolic lines, where r is the Witt index, and is denoted by U(2r, q^{2}). A unitary space of odd rank 2r + 1 over the field GF (q^{2}) is isometric to an orthogonal direct sum of r hyperbolic lines and an anisotropic subspace of rank 1, where r is the Witt index, and is denoted by U(2r + 1, q^{2}).

Lemma 1.9. (i) An anisotropic orthogonal space over GF (q) has rank at most 2. (ii) All anisotropic orthogonal spaces of rank 2 over GF (q) are isometric. (iii) If q is even, all anisotropic orthogonal spaces over GF(q) of rank 1 are isometric. (iv) If q is odd, all anisotropic orthogonal spaces over GF(q) of rank 1 are similar, but there are two isometry classes.

Proof : See Taylor (1992), pp138-139, or Cameron (year unknown), p76.

Corollary 1.3. An orthogonal space of even rank 2r over GF (q) is either isometric to an orthogonal direct sum of r hyperbolic lines, where r is the Witt index, and denoted by O^{+}(2r, q) and called hyperbolic, or is isometric to an orthogonal direct sum of r − 1 hyperbolic lines and an anisotropic space of rank 2, where r − 1 is the Witt index, and denoted by O^{−}(2r, q) and called elliptic. An orthogonal space of odd rank 2r + 1 over GF (q) is isometric to an orthogonal direct sum of r hyperbolic lines and an anisotropic space of rank 1, where r is the Witt index. If q is even, this determines the space up to isometry, while if q is odd, this determines the space up to similarity, but there are two isometry classes of such spaces. In any case we denote the space by O(2r + 1, q) and call it parabolic.

The two isometry classes in the odd rank orthogonal case for characteristic not 2 clearly give rise to the same polar space, however, care must be taken when such a polar space is embedded in a polar space of higher polar rank. In this case, a choice is made when considering the smaller polar space, as highlighted in the paper of Bader, Durante, Lunardon, Penttila and the present author ([8]). We will study this further in the next section, where discriminants are used to distinguish isometry classes of orthogonal spaces, and will also see this in the context of BLT-sets later.

The problem of axiomatising polar spaces was solved by Tits (1974) (building on work of Veldkamp (1959)).

An abstract polar space of polar rank r is a set of points together with a set of distinguished subsets called subspaces such that :

(i) any subspace, together with the subspaces it contains, is a projective space of dimension at most r − 1;

(ii) the intersection of any two subspaces is a subspace;

(iii) given a subspace H of dimension r − 1 and a point P not in H, there exists a unique subspace M containing P such that M ∩ H has dimension r − 2 and M contains all points of H collinear with P;

(iv) there exist two disjoint subspaces of dimension r − 1.

Axiom (iii) will be refered to as Tits’ polar space axiom in later sections.

These axioms were somewhat simplified by Buekenhout and Shult (1974), in the spirit of the Veblen-Young axioms for a projective geometry. They provided a set of axioms for an abstract polar space solely in terms of the points and lines of the space, from which all subspaces could be reconstructed. So for a set of points P together with a non-empty set of subsets called lines :

BS(i) every line has at least three points;

BS(ii) no point is collinear with all other points;

BS(iii) if a point P is not on a line l, then either P is collinear with exactly one point of l or P is collinear with all points of l.

A subset S of points is called a subspace if every pair of distinct points of S is collinear, and if whenever a line l contains two distinct points of S, all points of l are in S. Then if such a set of points P does not contain an infinite chain of distinct subspaces, Buekenhout and Shult showed that P together with its sets of subspaces satisfies the set of axioms given by Tits above. The polar rank is then one more than the size of a maximal subspace. Axiom BS(iii) will be refered to as the all-or-one axiom in later sections.

All projective spaces of dimension ≥ 3 are Desarguesian, whereas there exist projective planes which are non-Desarguesian. Similarly, a complete classification theorem of polar spaces of polar rank ≥ 3 was given by Tits (1974). For the case of finite polar spaces, this classification amounts to the following :

Theorem 1.6 (Tits (1974)). A finite polar space of polar rank ≥ 3 is a classical polar space.

Thus we have seen how to construct all finite polar spaces of polar rank ≥ 3. The polar spaces of rank 2, containing only points and lines, are called generalised quadrangles, and we will come to these later. Just as with projective planes, not all generalised quadrangles arise from polarities or from quadrics as the higher rank polar spaces do. (Rank 1 polar spaces are considered trivial structures.)

The six families of finite classical polar spaces can be summarised as in the following table. We use the more familiar geometric notation for these polar spaces in this thesis, so the dimensions in the table are projective dimensions, r is the polar rank, and e, depending only on the anisotropic space, carries information about all spaces in the family. (Note that for the unitary case, the finite field must have order a square, to ensure the existence of a nonidentity companion automorphism.)

Type  Notation  e
symplectic W (2r − 1, q)  0
unitary  H(2r − 1, q^{2}) −\frac{1}{2}
unitary H(2r, q^{2}) \frac{1}{2}
orthogonal Q^{+}(2r − 1, q) −1
orthogonal Q(2r, q)  0
orthogonal Q^{−}(2r + 1, q) 1

We list here some combinatorial results for finite polar spaces, proofs of which can be found, for example, in Cameron (year unknown) or Buekenhout (1995). The notation follows that of the above table.

Theorem 1.7. The number of points in a finite polar space of rank 1 is q^{1+e} + 1.

Theorem 1.8. In a finite polar space of rank r, there are (q^{r} − 1)(q^{r+e} + 1)/(q − 1) points, of which q^{2r−1+e} are not orthogonal to a given point.

Theorem 1.9. The number of maximal subspaces in a finite polar space of rank r is

∏_{i=1}^{r}(1 + q^{i+e}).

In particular, if r = 2 (so the polar space is a generalised quadrangle), each line has q + 1 points, and each point lies on q^{1+e} + 1 lines. In terms of the parameters we shall use later for generalised quadrangles, s = q and t = q^{1+e}.

The full collineation groups of the classical polar spaces are, in general, induced by the semisimilarity groups of the underlying spaces with form in the same manner as for projective spaces, that is, by factoring out the scalar maps. These groups are denoted as follows :

(i) symplectic : P Γ Sp(2r, q)

(ii) unitary : P Γ U(2r, q^{2}), P Γ U(2r + 1, q^{2})

(iii) orthogonal : P Γ O^{+}(2r, q), P Γ O(2r + 1, q), P Γ O^{−}(2r + 2, q).

The exceptional case is that of the polar space arising from the hyperbolic quadric Q^{+}(3, q) in PG(3, q). Note that this example is in fact a generalised quadrangle, and when considered as such there exist automorphisms permuting the lines in ways not possible when considered as a polar space for fields of order at least 4.

1.2 Orthogonal spaces and quadrics

The situation for orthogonal polar spaces is somewhat more involved than that for the other examples, and so we study them closer here.

Firstly, we justify the use of the superscripts + and − to distinguish isometry classes of orthogonal spaces of even rank, and introduce the concept of a discriminant when q is odd to distinguish the two isometry classes arising in general.

Consider the orthogonal direct sum of two nondegenerate orthogonal spaces U and W, each of even rank. If U and W are both hyperbolic, then U ⊥ W is hyperbolic by definition. If, without loss of generality, U is elliptic and W is hyperbolic, then U ⊥ W is elliptic by definition. However, if both U and W are elliptic, it is not so clear whether their orthogonal direct sum is hyperbolic or elliptic. Suppose V = U ⊥ W is elliptic. Since U is nondegenerate, V = U ⊥ U^{⊥}, and so by the classification above, W is isometric to U^{⊥}. But we know from the previous case that U^{⊥} must be hyperbolic when U and V are elliptic, and so U ⊥ W must be hyperbolic.

Thus we can determine the isometry type of an even rank orthogonal space if we can decompose the space into an orthogonal direct sum of nondegenerate even rank subspaces for which we know the isometry type of each.

For q odd, consider two nondegenerate orthogonal spaces V_{1}, V_{2} over GF (q) with quadratic forms Q_{1}, Q_{2} respectively, and suppose they are isometric with respect to some isometry T. Let X_{i} be a basis for V_{i} and let B_{i} be the nonsingular matrix of f_{Q_{i}} with respect to X_{i}, for i = 1, 2. Let P be the nonsingular transition matrix corresponding to T. Then B_{2} = P^{T} B_{1}P. So det(B_{2}) = det(P)^{2}.det(B_{1}). Thus, up to a nonzero square multiple, the determinants of the matrices of any bases of isometric orthogonal spaces are equivalent.

Let F^{∗} = GF (q)\{0} be the multiplicative group of GF (q), q odd. Then □ = {t^{2}|t ∈ F^{∗}} is a subgroup of F^{∗} of index 2. Let ⊠ be the other coset of □ in F^{∗}, so that F^{∗}/□ = {□, ⊠}.

Let V be a nondegenerate orthogonal space over GF (q), q odd, with quadratic form Q, and let B be the nonsingular matrix of some basis X of V with respect to the polar form f_{Q} of Q. Define the discriminant of Q as disc(Q) = det(B)□ ∈ F^{∗}/□. From the discussion above, it is clear that the discriminant is an invariant of nondegenerate orthogonal spaces over finite fields of odd order, up to isometry. In other words, such spaces are isometric if and only if they have the same discriminant.

Suppose V is the orthogonal direct sum of two nondegenerate orthogonal spaces U and W, with repsect to the quadratic form Q. Let A, B, C be matrices of V, U, W

respectively. Then A = (\begin{matrix}B & 0 \\ 0 & C \end{matrix}), and so disc(Q) = det(A)□ = (det(B)det(C))□ =

(det(B)□)(det(C)□) = disc(Q|_{U})disc(Q|_{W}), where Q|_{U} is the quadratic form obtained by restricting Q to the subspace U.

Motivated by this, the use of the superscripts + and − for the two isometry classes of nondegenerate orthogonal spaces of even rank is then familiar with the notion of multiplying positive and negative integers : “minus times minus is plus” corresponds to the orthogonal direct sum of two elliptic spaces being hyperbolic, as we saw earlier.

Let V be a nondegenerate orthogonal space of even rank 2r over GF (q), q odd, arising from the quadratic form Q. Suppose V is hyperbolic, that is, V = l_{1} ⊥

… ⊥ l_{r} for some hyperbolic lines l_{i}. Now l_{j} has matrix (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) with respect to f_{Q}

restricted to l_{j}. So disc(Q|_{l_{j}}) = (−1)□. Thus from above, disc(Q) = ((−1)□)^{r} = (−1)^{r}□.

Now −1 ∈ □ if and only if q ≡ 1(mod 4), so we have the following for a hyperbolic space :

disc(Q) = \begin{cases}□, if q ≡ 1(mod 4) or r is even \\ ⊠, if q ≡ 3(mod 4) and r is odd \end{cases}

So depending upon q and r, the value of the discriminant completely determines the isometry type of a nondegenerate orthogonal space of even rank 2r over a finite field of odd order; that is, the discriminant is a complete invariant of such spaces, up to isometry.

For nondegenerate orthogonal spaces of odd rank 2r +1, there are also two isometry classes. If, for q odd, we can show that both values of the discriminant do occur, then as the discriminant is an invariant up to isometry, it will be a complete invariant of such spaces, up to isometry.

So consider the following two nondegenerate quadratic forms over GF (q) :

Q_{1} : x_{0}x_{2r} + x_{1}x_{2r−1} + … + x_{r−1}x_{r+1} − x_{r}^{2};
Q_{n} : x_{0}x_{2r} + x_{1}x_{2r−1} + … + x_{r−1}x_{r+1} − nx_{r}^{2}.

Calculating the discriminant of each we have :

disc(Q_{1}) = \begin{cases}−1□, if r is even \\ 1□, if r is odd \end{cases} ; disc(Q_{n}) = \begin{cases}−n□, if r is even \\ n□, if r is odd \end{cases} .

Now if q ≡ 1 (mod 4), then −1 ∈ □, and so disc(Q_{1}) ∈ □ for all r, and disc(Q_{n}) ∈ ⊠ for all r.

If q ≡ 3 (mod 4), then −1 ∈ ⊠, and we can choose n = −1. Then

disc(Q_{1}) = \begin{cases}⊠, if r is even \\ □, if r is odd \end{cases} ; disc(Q_{n}) = \begin{cases}□, if r is even \\ ⊠, if r is odd \end{cases} .

So in general, the discriminant is a complete invariant of nondegenerate orthogonal spaces over finite fields of odd order, up to isometry.

Of course, for q even, □ = F^{∗}, and so the discriminant as defined has little meaning in this case.

We now study in more detail some quadrics of low-dimensional (Desarguesian) projective spaces, which will appear in many sections of material later.

From above, we know that all nondegenerate quadrics in PG(2, q) are similar. Such a quadric is called a conic. If the defining quadratic form is Q : x_{1}^{2} − x_{0}x_{2}, a conic may be parametrised as follows :

C = {⟨(t^{2}, t, 1)⟩|t ∈ GF (q)} ∪ {⟨(1, 0, 0)⟩}.

Thus a conic consists of q + 1 points. Our analysis of the discriminant for parabolic spaces shows that the other isometry class of conics may be represented by the quadratic form Q′ : nx_{1}^{2} − x_{0}x_{2}, for some nonsquare n ∈ ⊠.

Let l be a line of PG(n, q) and suppose l contains three distinct points P = ⟨u⟩, P′ = ⟨v⟩, R = ⟨u + av⟩ of some quadric Q, defined by the quadratic form Q. Then f_{Q}(u, v) = 0 (expand Q(u + av) = 0), and so for any point X = ⟨u + bv⟩ on l we have Q(u + bv) = 0. That is, if a line l contains three distinct points of the quadric Q, then l is a line of Q.

As the Witt index for a conic is 1, a conic contains no lines, and so no three points of a conic are collinear. This property motivates the general definition of a k-arc of a projective plane as a set of k points, no three collinear. The largest possible k-arc of a projective plane of order n has size n+2 and is called a hyperoval. Hyperovals exist only when n is even. A k-arc of size n + 1 is called an oval, and so a conic in PG(2, q) is an oval. For q odd, the converse is also true.

Theorem 1.10 (Segre (1955)). If q is odd, every oval in PG(2, q) is a conic.

For q even, there exist examples of ovals which are not conics. We also mention that a degenerate quadric in PG(2, q) is either a single point, a single line, or two intersecting lines, i.e., a subspace or the union of two hyperplanes.

Now consider PG(3, q). From above, we know there are two isometry classes of nondegenerate quadrics in PG(3, q), called hyperbolic and elliptic.

Let H be a nondegenerate hyperbolic quadric in PG(3, q). Then H has Witt index 2, and so contains both points and lines.

Let R be a nonempty set of mutually skew lines of PG(3, q), that is, no two distinct lines of R have a point in common. A line l is called a transversal of R if it meets each line of R in exactly one point. If through each point of each line of R there is a transversal of R, and through each point of a transversal of R there is a line of R, then we call R a regulus. The set R′ of all transversals of the regulus R forms a regulus, called the opposite regulus of R. Any regulus contains exactly q + 1 lines, and through any three distinct mutually skew lines, there is exactly one regulus.

The points of a nondegenerate hyperbolic quadric H in PG(3, q) are the points on some regulus R, and the lines of H are the lines of R together with the lines of the opposite regulus R′. This can easily be seen by considering the three mutually skew lines

l_{1} = ⟨(1, 0, 0, 0), (0, 1, 0, 0)⟩, l_{2} = ⟨(0, 0, 1, 0), (0, 0, 0, 1)⟩, l_{3} = ⟨(1, 0, 1, 0), (0, 1, 0, 1)⟩.

The points ⟨(x_{0}, x_{1}, x_{2}, x_{3})⟩ on the unique regulus through l_{1}, l_{2}, l_{3} satisfy the nondegenerate quadratic equation Q(x) = x_{0}x_{3} − x_{1}x_{2} = 0, which has disc(Q) = □.

The number of points on a nondegenerate hyperbolic quadric in PG(3, q) is then (q + 1)^{2}.

Just as the lines of H fall naturally into two classes, in general for hyperbolic spaces, the maximal totally singular subspaces form two distinct classes. Suppose that V is a hyperbolic orthogonal space of Witt index m > 0, and consider two maximal totally singular subspaces M, N of V. If we define a relation on the set of all maximal totally singular subspaces by considering for M and N the parity of the corank of their intersection in each, then we find that this defines an equivalence relation on the set of all maximal totally singular subspaces. There will be two equivalence classes, and subspaces are in the same class if and only if the corank of their intersection in each is even. It is clear that this relation is reflexive and symmetric, but the fact that it is also transitive is not so clear. See Cameron (year unknown) or Beutelspacher and Rosenbaum (1998) for a proof.

Now let E be a nondegenerate elliptic quadric in PG(3, q). Then E has Witt index 1, and so contains only points. Thus no three points of E are collinear.

Let p(x) ∈ GF (q)[x] be an irreducible polynomial of degree 2 over GF (q), and suppose p(x) = ax^{2} + bx + c. Define the nonconcurrent lines l = ⟨(1, 0, 0, 0), (0, 0, 0, 1)⟩ and m = {(0, x_{1}, x_{2}, 0)|ax_{1}^{2} +bx_{1}x_{2} +cx_{2}^{2} ≠ 0} of PG(3, q). Then, with respect to the quadratic form Q(x) = x_{0}x_{3} + ax_{1}^{2} + bx_{1}x_{2} + cx_{2}^{2}, l is a hyperbolic line and m is an anisotropic line. We know from the classification of such spaces above that E is an orthogonal direct sum of a hyperbolic line and an anisotropic line, and so the points ⟨x⟩ = ⟨(x_{0}, x_{1}, x_{2}, x_{3})⟩ of PG(3, q) such that Q(x) = 0 form an elliptic quadric E. The only point on E with x_{3} = 0 is ⟨(1, 0, 0, 0)⟩. If x_{3} ≠ 0, then, without loss of generality, take x_{3} = 1. Then x_{0} = −ax_{1}^{2} −bx_{1}x_{2} −cx_{2}^{2}, and so ⟨(−as^{2} −bst−ct^{2}, s, t, 1)⟩ is a point on E for all s, t ∈ GF (q). Hence the number of points on a nondegenerate elliptic quadric in PG(3, q) is q^{2} + 1.

A k-cap of PG(3, q) is a set of k points, no three collinear. The largest possible k-cap for q > 2 has size q^{2} + 1 and is called an ovoid. So an elliptic quadric in PG(3, q) is an ovoid. For q odd, the converse is also true.

Theorem 1.11 (Barlotti (1955), Panella (1955)). If q is odd, every ovoid in PG(3, q) is an elliptic quadric.

Now let C be a conic in PG(2, q), and embed PG(2, q) as a hyperplane π of PG(3, q). For a point P not on π, take the lines through P which meet C in a point. There are necessarily q + 1 such lines, and the set of (q^{2} + q + 1) points on these lines form a quadratic cone. The point P is called the vertex of the quadratic cone K, and K is said to project the conic C.

The pointwise stabiliser of π in P Γ L(4, q) is transitive on points not on π, and P Γ L(4, q)_{π} induces P Γ L(3, q) on π. Since all conics of PG(2, q) are equivalent under P Γ L(3, q), it follows that all quadratic cones of PG(3, q) are equivalent under P Γ L(4, q).

Suppose π is the plane x_{3} = 0 of PG(3, q), and take P = ⟨(0, 0, 0, 1)⟩ not on π. Consider the quadric Q of PG(3, q) defined by the quadratic form Q : x_{1}^{2}−x_{0}x_{2}. Then Q is the quadratic cone with vertex P meeting π in the conic C = {⟨(t^{2}, t, 1, 0)⟩|t ∈ GF (q)} ∪ {⟨(1, 0, 0, 0)⟩}. Clearly Q is degenerate, since the point P is collinear with all points of Q; in fact, the singular radical of Q is {(0, 0, 0, a) : a ∈ GF (q)} = P.

In PG(3, q) we also have the generalised notion of an oval cone, that is, a cone projecting an oval of some hyperplane of PG(3, q). Of course, for q odd, an oval cone is a quadratic cone, but for q even there exist examples of oval cones which are not quadratic cones. The other types of degenerate quadrics in PG(3, q) are a subspace or the union of two hyperplanes.

Note that restricting a quadratic form defining some quadric Q in PG(n, q) to a subspace S of PG(n, q) will define a quadric Q_{S} of S. If Q is an elliptic quadric, a hyperbolic quadric or a quadratic cone in PG(3, q), and π is a plane, then Q_{π} is a quadric of π. In the following chapters we will be interested in studying plane sections of these three quadrics which are conics.

1.2.1 The Klein correspondence The classical correspondence between projective 3-space and a nondegenerate hyperbolic quadric in projective 5-space is often referred to as the Klein correspondence, after the German mathematician Felix Klein (1849-1925), who first studied the correspondence over the complex numbers in his dissertation of 1868 (reprinted in his collected works). The correspondence is valid over an arbitrary field, and arises from defining a simple set of homogeneous coordinates for points in projective 5-space, called Plücker coordinates after the Prussian geometer and physicist Julius Plücker (1801-1868).

For a line l in PG(3, q) through the points ⟨u⟩ = ⟨(u_{0}, …, u_{3})⟩ and ⟨v⟩ = ⟨(v_{0}, …, v_{3})⟩, define a point P = ⟨(p_{01}(u, v), p_{02}(u, v), p_{03}(u, v), p_{12}(u, v), p_{31}(u, v), p_{23}(u, v))⟩ of PG(5, q), where p_{ij}(u, v) = u_{i}v_{j} − u_{j}v_{i} are the Plücker coordinates of l. Then P is a point on the nondegenerate hyperbolic quadric Q^{+}(5, q) of PG(5, q) defined by the quadratic form Q : x_{0}x_{5} + x_{1}x_{4} + x_{2}x_{3}. The bijective mapping from the set of all lines of PG(3, q) to the set of all points of Q^{+}(5, q) is often called the Klein mapping (or Plücker mapping) and Q^{+}(5, q) is called the Klein quadric. This mapping determines a complete correspondence between PG(3, q) and Q^{+}(5, q).

Concurrence of lines of PG(3, q) corresponds to collinearity (in a line of Q^{+}(5, q)) of points of Q^{+}(5, q), and so a maximal set of pairwise concurrent lines of PG(3, q) corresponds to a maximal set of pairwise collinear points of Q^{+}(5, q). A set of pairwise concurrent lines of PG(3, q) can be either a set of lines all through a common point, or a set of lines all in a common plane. A set of pairwise collinear points of Q^{+}(5, q) is a set of points on a maximal subspace of Q^{+}(5, q), that is, on a plane of Q^{+}(5, q). As Q^{+}(5, q) is a hyperbolic quadric, the planes fall into two classes, and these two classes correspond to the two possibilities for a maximal set of pairwise concurrent lines of PG(3, q), that is, to the points and the planes of PG(3, q). We label the planes of Q^{+}(5, q) corresponding to points of PG(3, q) as type I planes, and planes of Q^{+}(5, q) corresponding to planes of PG(3, q) as type II planes. Two planes of the same type intersect in a point, since two points (planes) of PG(3, q) lie on a unique line. Two planes of different types intersect in a line or are disjoint according to whether the corresponding point-plane pair of PG(3, q) is incident or nonincident, respectively. So the lines of Q^{+}(5, q) correspond to incident point-plane pairs of PG(3, q), and a line of Q^{+}(5, q) lies in one plane of type I and one plane of type II. Note that the arbitrary labelling of the planes of Q^{+}(5, q) is permitted, due to the principle of duality in projective spaces, and hence explains why the correspondence cannot be strengthened to an isomorphism of structures.

Consequently, the correlation group of PG(3, q) is isomorphic to P Γ O^{+}(6, q), the full collineation group of Q^{+}(5, q).

For a detailed list of substructures of PG(3, q) and Q^{+}(5, q) which Klein correspond, see Hirschfeld (1985).

Theorem 1.12. The dual of the polar space arising from the nondegenerate quadric Q(4, q) is isomorphic to the polar space arising from the nondegenerate symplectic space Sp(4, q).

Proof : Since all nondegenerate symplectic spaces of a given rank are isometric, we may take any alternating form as the underlying form on the space. The canonical form is f : x_{1}y_{2}−x_{2}y_{1}+x_{3}y_{4}−x_{4}y_{3}, so for all points ⟨x⟩, ⟨y⟩ on lines of the symplectic polar space Π(Sp(4, q)) we have

x_{1}y_{2} − x_{2}y_{1} + x_{3}y_{4} − x_{4}y_{3} = 0.

Thus the Plücker coordinates of the lines of Π(Sp(4, q)) satisfy p_{12} + p_{34} = 0. Now if we apply the Klein correspondence to these lines, the images will as usual lie on the Klein quadric, but they will now satisfy a quadratic equation in five variables, namely,

Q : −p_{12}^{2} + p_{13}p_{42} + p_{14}p_{23} = 0.

Thus the lines of Π(Sp(4, q)) correspond to the points of Π(Q(4, q)) defined by Q. All lines of Π(Sp(4, q)) on a point of Π(Sp(4, q)) are pairwise concurrent and so their images under the Klein correspondence are pairwise collinear in totally singular lines, and hence in the same totally singular line, since Q(4, q) does not contain planes. Hence the points of Π(Sp(4, q)) correspond to the lines of Π(Q(4, q)), and the polar spaces are dual to one another.

Theorem 1.13. The dual of the polar space arising from the nondegenerate quadric Q^{−}(5, q) is isomorphic to the polar space arising from the nondegenerate unitary space U(4, q^{2}).

Proof : Let τ ∈ P Γ O^{+}(6, q^{2}) be a collineation of Q^{+}(5, q^{2}) of order two, whose fixed points are the points of the subgeometry Q^{−}(5, q). (Such collineations exist, and are often called Baer involutions.) Then for any line l of Q^{+}(5, q^{2}) which meets Q^{−}(5, q) in a subline (fixed by τ), the plane of type I on l must be mapped by τ to the plane of type II on l. Hence τ interchanges the two classes of planes on Q^{+}(5, q^{2}), and so Klein corresponds to a duality of PG(3, q^{2}). Since τ has order two, the corresponding duality is a polarity π of PG(3, q^{2}). The number of points of Q^{−}(5, q) is (q+1)(q^{3}+1), by Theorem 1.8, and so τ fixes this number of points of Q^{+}(5, q^{2}). Hence π fixes (q + 1)(q^{3} + 1) totally isotropic lines of PG(3, q^{2}). Thus the polar rank of the polar space arising from π is at least two, and from the classification of such spaces in the previous section, it can be at most two. Hence, by Theorem 1.9, the polar space arising from π must be the polar space arising from the nondegenerate unitary space U(4, q^{2}). Thus the points of Π(Q^{−}(5, q)) correspond to the lines of Π(U(4, q^{2})), and so from the Klein correspondence, the lines of Π(Q^{−}(5, q)) correspond to the points of Π(U(4, q^{2})).

The above theorems can be found in Payne and Thas (1984). The second result appears in Thas and Payne (1976) and in Bruen and Hirschfeld (1978).

CHAPTER 2 Translation planes and elation generalised quadrangles

The previous chapter displays a strong analogy between projective spaces and polar spaces; especially with respect to the projective spaces of dimension 2 and the polar spaces of polar rank 2. The reason for this analogy is that these classes of “pointline” incidence structures can be viewed in a wider setting. We briefly discuss some of these ideas before focussing on the individual topics of interest.

An incidence structure I = (P, L, I) is a set of points P and a set of lines L, where P and L are disjoint, nonempty sets, and I ⊆ P × L defines incidence. Noting that the incidence graph of I is bipartite, we define a particularly nice class of incidence structures known as generalised polygons. A generalised n-gon is a bipartite graph of diameter n and girth 2n. To avoid trivial cases, we assume that n ≥ 3, and if the degree of each vertex is at least 3, a generalised n-gon is called thick. Then Feit and Higman (1964) proved that any finite, thick generalised n-gon has n ∈ {3, 4, 6, 8}. The generalised 3-gons are the projective planes, and the generalised 4-gons are the generalised quadrangles. For n = 6 they are called generalised hexagons, and for n = 8 they are called generalised octagons.

Generalised polygons were introduced by Tits (1959) while studying finite simple groups of Lie type in prime characteristic. They arose through the following observation.

Consider an incidence structure I = (P, L, I). An element (P, l) of I is called a flag. If a subgroup G of automorphisms of I acts transitively on the set I of flags, then G is said to be flag-transitive. Now suppose H and K are subgroups of some group G. Take P = (G : H) to be the set of cosets of H in G, L = (G : K) to be the set of cosets of K in G, and I = {(Hg, Kl) ⊆ P × L : Hg ∩ Kl ≠ ∅} to be nonempty intersection of cosets. Then I = (P, L, I) is an incidence structure on which the subgroup G of automorphisms of I acts flag-transitively. Tits (1956) observed the remarkable fact that every flag-transitive incidence structure is of this form, for some group G.

If G is a finite simple group of Lie type in characteristic p, p prime, then take U to be a Sylow p-subgroup of G, and let B = N_{G}(U) be the normaliser of U in G. Let M_{1}, …, M_{r} be the maximal subgroups of G containing B. Then r is the Lie rank of G, and if r = 2, the incidence structure with points (G : M_{1}), lines (G : M_{2}) and incidence as nonempty intersection is a flag-transitive generalised polygon. The generalised polygons arising are essentially the classical examples within each class; that is, for example, the generalised 3-gons arising (from P SL(3, q)) are the Desarguesian projective planes PG(2, q), and the generalised 4-gons arising are (essentially) the finite classical polar spaces of polar rank 2. (If r > 2 we (essentially) obtain the finite Desarguesian projective spaces of dimension r and the finite classical polar spaces of polar rank r.)

The aim of this chapter is to introduce the study of two classes of generalised polygons which are intimately connected with our central topic, namely, translation projective planes and elation generalised quadrangles. In some sense, we weaken the concept of flag-transitivity in the above coset geometry construction, and instead act regularly on the elements furthest from some fixed element in the incidence graph of the generalised polygon. The somewhat natural approach for the projective planes motivates the analogous concepts for the generalised quadrangles. Furthermore, translation projective planes and elation generalised quadrangles represent most known examples of these two types of generalised polygons. For a complete study of all generalised polygons see, for example, van Maldeghem (1998). For recent surveys on topics in the theory of generalised polygons, see [42].

As many examples of these incidence structures are not naturally embedded in projective spaces, we cannot consider equivalence of such structures under the automorphism group of some projective geometry. As mentioned earlier, mathematical equivalence is considered to be structure-preserving, and so two point-line incidence structures I_{1} = (P_{1}, L_{1}, I_{1}) and I_{2} = (P_{2}, L_{2}, I_{2}) are said to be isomorphic if and only if there exist bijective mappings from P_{1} onto P_{2}, L_{1} onto L_{2} and I_{1} onto I_{2}.

2.1 Projective planes

We begin with a short discussion of projective and affine planes. From the axioms for a projective space given earlier, a projective plane is an incidence structure consisting of a set of points and a set of distinguished subsets called lines, which satisfies the following three conditions :

(i) any two points lie on a unique line;

(ii) any line contains at least three points;

(iii) any two lines meet in a unique point.

Recall that the order of a projective plane is defined to be one less than the number of points on a line, which we saw earlier is a constant for any projective geometry.

Thus the order of a projective plane is at least 2.

An affine plane is a point-line incidence structure satisfying the following three conditions :

(i) any two points lie on a unique line;

(ii) if l is a line and P is a point, then there is a unique line containing P and parallel to (i.e., equal to or disjoint from) l;

(iii) there exist three noncollinear points.

Note that the first two conditions imply that parallelism is an equivalence relation on the set of lines, and that each equivalence class partitions the set of points. The number of points on a line of an affine plane is the order of the affine plane.

The connection between projective and affine planes is simple : by removing a line l and all of its points from a projective plane P, one obtains an affine plane A = P^{l}. Conversely, given an affine plane, adjoin a ‘point at infinity’ corresponding to each parallel class of lines, which is incident with every line of the class, and adjoin a ‘line at infinity’ incident with all points at infinity. The structure obtained is a projective plane, sometimes called the projective completion of the affine plane. Hence an affine plane is equivalent to a projective plane with a distinguished line.

This concept naturally generalises to projective and affine spaces, where we remove a hyperplane and all of its subspaces from a projective space to obtain an affine space.

For a Desarguesian projective space we have a natural coordinate system arising from the choice of some basis for the underlying vector space. Since points of the projective space are represented by vector subspaces of rank one, these coordinates are homogeneous in that they are defined up to a scalar multiple. For non-Desarguesian projective planes, coordinates can be introduced by fixing an (ordered) quadrangle, that is, a set of four points, no three collinear. With respect to this quadrangle, coordinates can be defined in the affine plane arising from some line of the quadrangle, by setting up a one-to-one correspondence between the points of the affine plane and the ordered pairs of elements of some set T of cardinality equal to the order of the affine plane. A ternary operation can be defined on such a set T representing an arbitrary line of the affine plane, and satisfying certain properties arising from the axioms for the projective plane. Such a set T is called a ternary field (or planar ternary ring), and from any such ternary field, an affine plane may be constructed, so that there is a canonical correspondence between ternary fields and projective planes with distinguished ordered quadrangles. If the ternary operation is written as a · b ∘ c for a, b, c ∈ T , then addition in T is defined as a + b = a · 1 ∘ b and multiplication as ab = a·b∘0, where 0 and 1 are two distinct elements of T . Coordinates for points can then be given in the forms (0, 0, 1), (0, 1, a), (1, a, b) for a, b ∈ T and coordinates for lines in the forms [0, 0, 1], [0, 1, a], [1, a, b] for a, b ∈ T . These homogeneous coordinates have the property that the point (x_{0}, x_{1}, x_{2}) is incident with the line [y_{0}, y_{1}, y_{2}] if and only if y_{0}x_{2} = x_{1} · y_{1} ∘ x_{0}y_{2}. The projective plane is Desarguesian if and only if the ternary field is a skew field, and so in the finite case, as we have seen, if and only if the ternary field is GF (q). For more details and precise definitions of ternary fields, see Dembowski (1968), pp127-131. A good reference for the theory of projective planes in general is Hughes and Piper (1973).

2.1.1 Translation planes A collineation of a projective geometry was defined earlier. For an affine geometry, it is the same concept, that is, a permutation of the points which maps lines onto lines, and hence maps subspaces onto subspaces.

A point P is a centre of the collineation ϕ if all subspaces containing P are fixed by ϕ. A hyperplane H is an axis of ϕ if every point of H is fixed by ϕ. We call a collineation central or axial if it has a centre or an axis, respectively. Nontrivial collineations can have at most one centre and at most one axis, and a collineation of a projective geometry has a centre if and only if it has an axis. A central collineation is called an elation or a homology if it has an incident or nonincident centre-axis pair, respectively.

Let A be an affine plane and P the corresponding projective plane, so that A = P^{l} for some line l of P. A collineation of A is called a translation if, when considered as a collineation of P, it is an elation with axis l. The set of all translations of an affine plane A forms a group T under composition, and we call A a translation affine plane if T is transitive on the points of A. The projective completion P of A is then called a translation plane. We may rephrase this by saying that a projective plane π is a translation plane if and only if there is a line l of π and a subgroup G of Autπ, which fixes every point on l, and acts regularly on the set of points not on l. The group G is called the translation group, and l the translation line.

Let A be a translation affine plane and T be its full translation group. Suppose A = P^{l} and consider T as the group of all elations with axis l of the projective plane P. If we define G(P, l) to be the set of all collineations of P with centre P and axis l, then G(P, l) is a subgroup of the full collineation group of P, and so T = ⋃_{P Il} G(P, l). As the P Il are just the parallel classes of A, we can write T = ⋃_{i} T_{i}, where T_{i} denotes the subgroup of T which fixes every line of the i-th parallel class of A. Since the centre and axis of a nontrivial collineation are unique, it follows that T_{i}∩T_{j} = 1 for i ≠ j, otherwise there would exist a nontrivial collineation fixing two distinct parallel classes linewise, and thus fix linewise two distinct points of l. Thus the T_{i} form a partition of T, and furthermore, this is a congruence partition of T in the sense that if i ≠ j then T_{i}T_{j} = T.

Conversely, if T is an abstract group possessing a nontrivial congruence partition, i.e., a set of subgroups T_{i} such that T = ⋃_{i} T_{i}, T_{i} ∩ T_{j} = 1 and T_{i}T_{j} = T for i ≠ j, then T may be regarded as the full translation group of a translation affine plane. The incidence structure with points the elements of T, with lines the cosets of the T_{i} in T, and with incidence as set theoretic inclusion, is a translation affine plane with T as its full translation group.

Hence there is a canonical correspondence between translation affine planes and congruence partitions of groups, that is, the isomorphism classes of translation affine planes can be put into a one-to-one correspondence with the isomorphism classes of groups with congruence partitions. This was first proved by André (1954).

The kernel of the translation affine plane A is the set K(A) of all endomorphisms α of the translation group T such that T_{i}^{α} ⊆ T_{i} for all i parallel classes. With usual addition and multiplication of endomorphisms, K(A) is a ring, and in fact André (1954) showed it is also a skew field, and hence in the finite case, GF (q). As translations commute with one another, the translation group T of a translation affine plane is abelian, and in fact, T is elementary abelian. Thus regarding T as a vector space over some finite field F ⊆ K(A), we see that the subgroups T_{i} of the congruence partition are in fact subspaces of T. Hence in the corresponding projective geometry PG(T) arising from the vector space T, these subgroups form a collection of mutually disjoint subspaces covering all of PG(T). Such a collection S is called a spread of PG(T), and furthermore, S has the property that any two distinct subspaces of S span the whole of PG(T). Conversely, every spread of PG(V) with this property defines a congruence partition of the additive group of the underlying vector space V, and hence defines a translation affine plane, as above. Thus we have in the finite case :

Theorem 2.1 (André (1954)). For a finite field F of order q and positive integer t, there is a canonical correspondence between the translation affine planes A of order q^{t+1} with F ⊆ K(A), and the t-dimensional subspace spreads of PG(2t+1, q).

This was first proved by André (1954) in more generality, and then later rediscovered independently by Bruck and Bose (1964) and by Segre (1964). In fact, most known finite projective planes are either translation planes or closely related to them. See Kallaher (1982), Lüneburg (1980) and Ostrom (1970) for more on translation planes.

The coordinatising ternary field of a translation affine plane is known as a quasifield. As an example, we will consider a certain type of quasifield known as a semifield. Let S be a finite set, with two binary operations “+” and “·” such that

(i) S(+) is an abelian group, with identity 0;

(ii) both distributive laws hold in S;

(iii) there exists a multiplicative identity 1;

(iv) x · y = 0 if and only if x = 0 or y = 0.

Then S is called a (finite) semifield, and if multiplication is not associative, so that S is not a field, S is called a proper semifield. The middle nucleus N = {n ∈ S : (xn)y = x(ny), ∀x, y ∈ S} of S is a field, and S may be regarded as a vector space over N. Clearly, a proper semifield has N ≠ S. A finite semifield necessarily has prime power order, and proper finite semifields exist for all orders q = p^{h}, where h ≥ 2 if p is odd and h ≥ 4 if p = 2. A translation plane coordinatised by a semifield is called a semifield translation plane.

Let S be a (multiplicatively) commutative semifield of order q^{2} whose middle nucleus is GF (q), so that S may be regarded as a vector space of rank 2 over GF (q). Then S = {tα + β : α, β ∈ GF (q)} for some t ∈ S\GF (q). Let addition in S be component-wise, and multiplication be defined as follows :

(tα + β)(tγ + δ) = t(t(αγ)) + t(βγ) + t(αδ) + βδ.

If we write t(tx)) = tf(x) + g(x) for all x ∈ GF (q), where f, g : GF (q) → GF (q), then we have :

(tα + β)(tγ + δ) = t[f(αγ) + βγ + αδ] + [g(αγ) + βδ].

With respect to this form of the multiplication, considering both distributive laws and the existence of zero divisors yields the following result.

Lemma 2.1 (Cohen and Ganley (1982)). The above multiplication involving the maps f and g yields a semifield if and only if

(i) f and g are linear maps from GF (q) to GF (q), and

(ii) xy^{2} + f(x)y − g(x) ≠ 0 for all x, y ∈ GF (q), with x ≠ 0.

The maps f(x) = αx and g(x) = βx, where α, β ∈ GF (q) and w^{2} + αw − β is irreducible over GF (q), yield an improper finite semifield of order q^{2}, namely GF (q^{2}). Surprisingly, we have the following for p = 2.

Theorem 2.2 (Cohen and Ganley (1982)). There is no proper commutative semifield of even order which is of rank 2 over its middle nucleus.

For q odd however, there exist examples of proper commutative semifields of rank 2 over their middle nuclei.

Example (The Ganley semifields [Ganley (1981)]) Let q = 3^{h}, h ≥ 2, and fix a nonsquare n of GF (q). Define the following linear maps from GF (q) to GF (q) :

f(x) = nx^{3} and g(x) = nx^{9} + n^{3}x.

To check the second condition in the above lemma, we must show that f^{2}(x)+4xg(x) is a nonsquare of GF (q) for all x ≠ 0. Then

f^{2}(x) + 4xg(x)
= n^{2}x^{6} + nx^{10} + n^{3}x^{2}
= nx^{2}(nx^{4} + x^{8} + n^{2})
= nx^{2}(x^{4} − n)^{2}

and we have a proper commutative semifield of order q^{2}.

2.2 Generalised quadrangles

The finite classical polar spaces of polar rank 2 consist of points and lines. Each line has q + 1 points, and each point lies on q^{1+e} + 1 lines, where e is as in the table of finite classical polar spaces earlier.

Consider also Tits’ polar space axiom from earlier. For polar spaces of polar rank 2 this says the following : given a line l and a point P not on l, there exists a unique subspace M containing P such that M ∩ l is a point and M contains all points of l collinear with P. So M must be a line, and with respect to the all-or-one axiom of Buekenhout and Shult, this says that for polar spaces of rank 2, the ‘all’ alternative does not occur.

A generalised quadrangle is an incidence structure of points and lines with a symmetric incidence relation I satisfying

(i) each point is incident with a constant number t + 1 of lines, and two distinct points are incident with ≤ 1 line;

(ii) each line is incident with a constant number s + 1 of points, and two distinct lines are incident with ≤ 1 point;

(iii) for a non-incident point-line pair (P, l) there is a unique point Q and a unique line m such that P I m I Q I l.

We will often abbreviate generalised quadrangle as GQ. We call (s, t) the order of a GQ, and if s = t, we say the GQ has order s. For a GQ X of order (s, t), the incidence structure with points the lines of X and lines the points of X is a GQ of order (t, s), called the dual GQ of X. If s > 1 and t > 1, a GQ of order (s, t) is called thick; otherwise thin, and is an example of a grid (if t = 1) or a dual grid (if s = 1).

For a GQ X = (P, L, I), we often write P ∼ Q for P, Q ∈ P if P and Q are collinear (in a line of X); P^{⊥} for {Q ∈ P|P ∼ Q}; {P, Q}^{⊥} for P^{⊥} ∩ Q^{⊥}; {P, Q}^{⊥⊥} for ∩_{R∈{P,Q}^{⊥}}R^{⊥}; and use the obvious dual notation.

Fix a line l of a GQ X of order (s, t). Each point P of X not on l lies on a unique line m concurrent with l, say in the point Q. There are s + 1 points Q on l, each lying on t lines m ≠ l, each containing s points P ≠ Q, so the number of points not on l is (s+1)st. Hence the number of points of a GQ of order (s, t) is (s+1)(st+1). The dual of a GQ of order (s, t) is a GQ of order (t, s), and so the number of lines of a GQ of order (s, t) is (t + 1)(st + 1).

Theorem 2.3 (Higman (1974)). If X is a GQ of order (s, t) with s > 1 and t > 1, then s ≤ t^{2}, and dually, t ≤ s^{2}.

For these and many more results concerning finite generalised quadrangles we refer to the detailed monograph of Payne and Thas (1984).

2.2.1 The classical generalised quadrangles We list here the generalised quadrangles arising as classical polar spaces of polar rank 2.

1. W(q)=W(3, q)=Π(ρ) where ρ is a null polarity of PG(3, q) is a GQ of order q.

2. Q(3, q)=Q^{+}(3, q) = Π(Q^{+}) where Q^{+} is a hyperbolic quadric of PG(3, q) is a GQ of order (q, 1).

3. Q(4, q)=Π(Q) where Q is a parabolic quadric of PG(4, q) is a GQ of order q.

4. Q(5, q)=Q^{−}(5, q)=Π(Q^{−}) where Q^{−} is an elliptic quadric of PG(5, q) is a GQ of order (q, q^{2}).

5. H(3, q^{2})=Π(ρ) where ρ is a unitary polarity of PG(3, q^{2}) is a GQ of order (q^{2}, q).

6. H(4, q^{2})=Π(ρ) where ρ is a unitary polarity of PG(4, q^{2}) is a GQ of order (q^{2}, q^{3}).

The GQ Q(3, q) of order (q, 1) is a thin GQ (or a grid), which is often considered a trivial structure, though being a hyperbolic quadric in PG(3, q), often plays a very nontrivial role in much beautiful mathematics.

Recall from earlier the duality between W(q) and Q(4, q) (Theorem 1.12), and the duality between Q(5, q) and H(3, q^{2}) (Theorem 1.13). Both of these pairs of dual structures will be very important in the following chapters.

The GQ H(4, q^{2}) will not play a role in this thesis.

2.2.2 The constructions of Tits Some of the first examples of nonclassical GQs were given by Tits (and first appeared in Dembowski (1968)), presumably generalising the classical examples arising from quadrics.

Observe that by fixing a point P of a GQ, we can partition both the set of points other than P and the set of lines into subsets determined by their incidence relationship with P. That is, the points other than P are either collinear or noncollinear with P, and the lines are either incident or nonincident with P. This can be represented by the following diagram :

lines  points  lines  points
P on P collinear not on P not collinear
with P  with P
•  ⨀  ⨀  ⨀  ⨀

1. T_{2}(O) - generalising Q(4, q)

Let P be a point of Q(4, q). Project from P onto a hyperplane PG(3, q) not on P. Then P^{⊥}∩ PG(3, q) is a plane π. The lines through P project a conic C of π. Every line through P not in P^{⊥} is secant to the parabolic quadric Q defining Q(4, q), so the points not collinear with P project the whole of PG(3, q)\π. The lines not on P project to lines l of PG(3, q) with l ∩ π a point of C. The points collinear with P are in one-to-one correspondence with their tangent spaces, which project to planes π′ meeting π in a tangent line to C.

This motivates the following construction, as a conic is an oval.

Let O be an oval of PG(2, q). Embed PG(2, q) as a plane π of PG(3, q). Define the incidence structure T_{2}(O) by :

lines  points  lines  points
(∞) on (∞)  collinear  not on (∞) not collinear
with (∞)  with (∞)
• ⨀ ⨀ ⨀ ⨀
planes  lines
(∞) points  meeting π  meeting π  points
of O  in a tangent in a point  not on π
line to O  of O

with (∞) on all points of O, and all other incidences inherited from PG(3, q).

Theorem 2.4 (Tits (unpublished, first appeared in Dembowski (1968))). T_{2}(O) is a GQ of order q. T_{2}(O) ≅ Q(4, q) ⇔ O is a conic.

2. T_{3}(Ω) - generalising Q(5, q)

If we repeat our analysis of Q(4, q) working out from a point with Q(5, q), since a tangent hyperplane to an elliptic quadric in PG(5, q) meets the quadric in a cone projecting an elliptic quadric of PG(3, q), and an elliptic quadric is an ovoid, we are led to the following :

Let Ω be an ovoid of PG(3, q). Embed PG(3, q) as a hyperplane Σ of PG(4, q). Define an incidence structure T_{3}(Ω) with special point (∞), lines on (∞) being points of Ω, points collinear with (∞) being hyperplanes of PG(4, q) meeting Σ in a tangent plane to Ω, lines not on (∞) being lines of PG(4, q) meeting Σ in a point of Ω, and points not collinear with (∞) being points of PG(4, q) not on Σ.

lines  points  lines  points
(∞) on (∞)  collinear  not on (∞) not collinear
with (∞)  with (∞)
• ⨀ ⨀ ⨀ ⨀
hyperplanes  lines
(∞) points  meeting Σ  meeting Σ  points
of Ω  in a tangent in a point  not on Σ
plane to Ω  of Ω

Theorem 2.5 (Tits (unpublished, first appeared in Dembowski (1968))). T_{3}(Ω) is a GQ of order (q, q^{2}). T_{3}(Ω) is isomorphic to Q(5, q) ⇔ Ω is an elliptic quadric.

2.2.3 Elation generalised quadrangles A collineation of a generalised quadrangle is a permutation of the points which maps lines onto lines. A collineation fixing every line incident with some point P is called a whorl about P. A whorl about P, which is either the identity collineation or which fixes no point not collinear with P, is called an elation about P. Motivated by the study of translation planes, we consider those GQ for which there exists a set of elations about some point P which forms a group acting transitively on the set of points not collinear with P.

A generalised quadrangle X is called an elation generalised quadrangle (EGQ) if for some point P of X there is a subgroup G of the automorphism group AutX of X which fixes each line on P and acts regularly on the set of points not collinear with P. The group G is called the elation group, and P the base point.

Example (T_{2}(O) is an EGQ) The set G of all elations of PG(3, q) with axis the plane π is a group which acts regularly on the set of all points of PG(3, q) not on π. G also fixes every point of O, and acts naturally on the set of planes meeting π in a tangent line to O, and on the set of lines meeting π in a point of O. By defining (∞)g = (∞) ∀g ∈ G, we identify G with a subgroup of AutT_{2}(O) which fixes (∞), every line of T_{2}(O) on (∞) and acts regularly on the set of points of T_{2}(O) not collinear with (∞).

We saw earlier that given a particular system of subgroups of an abstract group T, namely a congruence partition of T, it is possible to define an incidence structure in terms of cosets of these subgroups which is a translation affine plane, having T as its full translation group. A similar approach via a group coset geometry also exists for EGQs.

Given an EGQ X of order (s, t) with an elation group G and base point P, fix a point Q, not collinear with P. Then for every point R not collinear with P, there is a unique element g of G with Qg = R, since G acts regularly on the set of points not collinear with P. Identifying R with g, we see that the points not collinear with P can be identified with the elements of G (with Q being the identity of G). As there are s^{2}t points not collinear with P, G must have size s^{2}t.

Now consider a point P′ collinear with P. There is a unique line l on P and P′ and ∀g ∈ G lg = l. Let P′′ be a point of l with P ≠ P′′. There exist points Q′ and Q′′, not collinear with P, with Q′ collinear with P′ and Q′′ collinear with P′′. So there exists g ∈ G with Q′g = Q′′. Since lg = l, and P′ is the unique point on l collinear with Q′, and P′′ is the unique point on l collinear with Q′′, it follows that P′g = P′′. So G acts transitively on the set of points on l, other than P. There are s of these points, so, by the orbit-stabiliser theorem, the stabiliser G_{P′} of P′ in G has order st. The points on l, other than P, can then be identified with the cosets G_{P′}g of G_{P′} in G (g ∈ G). So to reconstruct in G all of the points collinear with P, we will need one point P′_{i} on each of the t + 1 lines l_{i} on P, giving rise to t + 1 subgroups S_{0}^{∗}, …, S_{t}^{∗} of G, each of order st. Now the points collinear with P can be identified with the cosets S_{i}^{∗}g, g ∈ G, i ∈ {0, …, t}. We choose P′_{i} to be the point on l_{i} collinear with Q.

Now consider a line l′ not on P. There is a unique line l and a unique point P′ with P incident with l incident with P′ incident with l′. Consider another line l′′ not on P, but concurrent with l. Then there are points Q′, Q′′ not collinear with P with Q′ on l′ and Q′′ on l′′. So there exists g ∈ G with Q′g = Q′′. Since l is on P, lg = l, and so l′g = l′′, as l′ is the unique line on Q′ concurrent with l and l′′ is the unique line on Q′′ concurrent with l. So G acts transitively on the set of lines concurrent with l, but not on P. There are st of these lines, so, by the orbit-stabiliser theorem, the stabiliser G_{l′} of l′ in G has order s. The lines concurrent with l, but not on P, can then be identified with the cosets G_{l′}g of G_{l′} in G (g ∈ G). So to reconstruct in G all of the lines not on P, we will need one line l′_{i} concurrent with each of the t + 1 lines l_{i} on P, giving rise to t + 1 subgroups S_{0}, …, S_{t} of G, each of order s. Now the lines not on P can be identified with the cosets S_{i}g, g ∈ G, i ∈ {0, …, t}. We choose l′_{i} incident with P′_{i}, so that the stabiliser S_{i} of l′_{i} fixes P′_{i}, that is, S_{i} ≤ S_{i}^{∗}, for all i ∈ {0, …, t}. We also choose l′_{i} to be incident with Q.

Now we have a group G of order s^{2}t with t + 1 subgroups S_{0}, …, S_{t} of order s and t + 1 subgroups S_{0}^{∗}, …, S_{t}^{∗} of order st such that S_{i} ≤ S_{i}^{∗}. Let J = {S_{0}, …, S_{t}} and J^{∗} = {S_{0}^{∗}, …, S_{t}^{∗}}.

We define the incidence structure S(G, J , J^{∗}) as follows :

Points : (iii) a symbol (∞) (ii) cosets S_{i}^{∗}g (g ∈ G, i ∈ {0, …, t})

(i) elements of G

Lines : (b) symbols [S_{i}], i ∈ {0, …, t})

(a) cosets S_{i}g (g ∈ G, i ∈ {0, …, t})

The point (∞) is incident with all lines [S_{i}] of type (b). A line [S_{i}] of type (b) is incident with (∞) and with all points S_{i}^{∗}g of type (ii) with the same i. A point S_{i}^{∗}g of type (ii) is incident with the line [S_{i}] of type (b) with the same i and with each line S_{i}h of type (a) with the same i such that S_{i}h ⊆ S_{i}^{∗}g. A line S_{i}h of type (a) is incident with the point S_{i}^{∗}g of type (ii) with the same i such that S_{i}h ⊆ S_{i}^{∗}g and with each point g′ ∈ G of type (i) such that g′ ∈ S_{i}h. A point g of type (i) is incident with each line S_{i}g of type (a).

Note that we use the same labelling of the points and lines of S(G, J , J^{∗}) which has become standard, but reverse the order of presentation to emphasise that we have constructed S(G, J , J^{∗}) by working out from a point in the incidence graph of S(G, J , J^{∗}).

If X is an EGQ of order (s, t) with elation group G and base point P, l_{0}, …, l_{t} are lines of X on P, P′_{i} is a point on l_{i} (≠ P) for i ∈ {0, …, t}, l′_{i} is a line on P′_{i} (≠ l_{i}) for i ∈ {0, …, t}, S_{i} = G_{l′_{i}}, S_{i}^{∗} = G_{P′_{i}}, J = {S_{0}, …, S_{t}} and J^{∗} = {S_{0}^{∗}, …, S_{t}^{∗}}, then X is isomorphic to S(G, J , J^{∗}).

However, if we don’t construct subgroups of G in this way from a known GQ, what conditions on J and J^{∗} are necessary for S(G, J , J^{∗}) to be a GQ ? The following example illustrates the answer.

Example (T_{2}(O) as a group coset geometry) The elation group G of T_{2}(O) can be naturally identified with the vector space GF (q)^{3}, considered as an additive group. We can also think of this as AG(3, q).

The plane at infinity of AG(3, q) is identified with the plane π. We identify Q with the origin O(0, 0, 0) (i.e., with the identity of G). Now lines of AG(3, q) through Q correspond to 1-dimensional subspaces of GF (q)^{3}, and so to points of π. Thus l′_{0}, …, l′_{q}, the lines through Q, correspond to S_{0}, …, S_{q}, each of which is a point of π. Also, l′_{i}∩π is a point of O. So J is O. Now planes of AG(3, q) through Q correspond to 2-dimensional subspaces of GF (q)^{3}, and so to lines of π. Thus P′_{0}, …, P′_{q}, the points collinear with Q and collinear with (∞), correspond to S_{0}^{∗}, …, S_{q}^{∗}, each of which is a line of π. Also, these planes of PG(3, q) meet π in a tangent line to O. So J^{∗} is O^{∗}, the set of all tangent lines to O.

Now no three points of O are collinear, so P Q ∩ R = {0} for distinct P, Q, R in O. Thus

(K1) S_{i}S_{j} ∩ S_{k} = 1

for distinct i, j, k, where S_{i}S_{j} = {gh|g ∈ S_{i}, h ∈ S_{j}}.

Also, no point P of O lies on a tangent line to another point Q of O, i.e., ∀P_{i} ∈ O ∀l_{j} ∈ O^{∗} i ≠ j ⇒ P_{i} ∩ l_{j} = {0}, giving

(K2) S_{i} ∩ S_{j}^{∗} = {1}

for distinct i, j.

A system J, J^{∗} of subgroups is called a Kantor (4-gonal) family if it satisfies the conditions (K1) and (K2). This construction of EGQs is originally due to Kantor (1980).

Theorem 2.6 (Kantor (1980)). S(G, J , J^{∗}) is a GQ if and only if J, J^{∗} is a 4-gonal family. In this case, S(G, J , J^{∗}) is an EGQ of order (s, t) with elation group G and base point (∞).

We remark that there exist examples of generalised quadrangles which do not occur in any of the models we have thus far considered. These examples, due first to Ahrens and Szekeres (1969) for all prime powers q and independently to Hall Jr. (1971) for q even, but later included in a more general construction by Payne (1971), are of order (q − 1, q + 1). They are the only known examples, with s > 1 and t > 1, in which s and t are not powers of the same prime.

CHAPTER 3 Flocks of circle planes

We will now focus on quadrics of PG(3, q), in particular, on elliptic quadrics, hyperbolic quadrics, and quadratic cones. These three families of quadrics are examples of certain incidence structures known as circle planes, namely inversive (or Möbius) planes, Minkowski planes and Laguerre planes, respectively. Indeed, each of these classes of circle planes, or Benz planes as they are also known, is a generalisation of the model of the plane sections of the quadric which are conics. The three families of quadrics are known as the Miquelian circle planes, in that they satisfy the theorem of Miquel, an incidence theoretic result somewhat analogous to the theorem of Pappus in projective planes. The generalised concepts of ovoid and oval cone in PG(3, q) also give rise to circle planes, modelling the plane sections which are ovals, and these examples are not Miquelian (unless, of course, the ovoid is an elliptic quadric or the oval cone is a quadratic cone).

The general theory of circle planes is very beautiful, however we do not have the time to delve into it here. Instead, we have mentioned this wider setting in order to introduce historically the main problem of this thesis. Circle planes are incidence structures arising from a set of points, a set of subsets of points called circles, and either 0,2 or 1 sets of equivalence classes of points called lines, for inversive, Minkowski or Laguerre planes respectively. For an inversive plane I, Dembowski (1968) introduced the concept of a flock as a set F of mutually disjoint circles such that, with the exception of precisely two points, every point of I is on a (necessarily unique) circle of F. In other words, a flock of I is a partition of all but two points of I into pairwise disjoint circles. This concept naturally extends to the other circle planes, and has generated a great deal of interest in many parts of geometry.

This thesis is concerned with the problem of flocks of the Miquelian circle planes, i.e., with flocks of elliptic quadrics, hyperbolic quadrics, and quadratic cones of PG(3, q). We discuss these separately here.

For the case of elliptic quadrics, we may discuss the more general situation of flocks of ovoids of PG(3, q). A flock of an ovoid of PG(3, q) is a partition of all but two points of the ovoid into pairwise disjoint ovals, and we speak of the planes of a flock as the set of planes containing the ovals. Then two planes of a flock must meet in a line external to the ovoid. If all planes of a flock meet in the same external line, we call the flock linear.

Theorem 3.1 (Orr (1973) (q odd), Thas (1973) (q even)). Any flock of an ovoid of PG(3, q) is linear.

A flock of a hyperbolic quadric of PG(3, q) is a partition of the points of the quadric into pairwise disjoint conics. Again, we speak of the planes of a flock as the set of planes containing the conics. Two planes of a flock must meet in an external line, and if all planes of a flock meet in the same external line, we call the flock linear.

Theorem 3.2 (Thas (1975)). Any flock of a hyperbolic quadric of PG(3, q), q even, is linear.

For hyperbolic quadrics over fields of odd order, there exist examples of flocks which are not linear.

Example (The Thas flocks [Thas (1975)]) Let q be odd, V be GF (q)^{4} and Q : V → GF (q) be a nondegenerate quadratic form of plus type, so that Q^{+} = {⟨v⟩ : Q(v) = 0} is a hyperbolic quadric of PV = PG(3, q). Let l be an external line to Q^{+} and m be the polar line to l, so that m = l^{⊥} = {v ∈ V |f(v, u) = 0 ∀u ∈ l} where f(x, y) = Q(x + y) − Q(x) − Q(y) is the polar form of Q. Let

F = {⟨v⟩^{⊥}|v ∈ l and Q(v)□ = □ or v ∈ m and Q(v)□ = −⊠}.

Then F is a flock of Q, which is called a Thas flock. (This follows from the fact that the discriminant of Q restricted to ⟨u, v⟩ for ⟨u⟩^{⊥}, ⟨v⟩^{⊥} ∈ F is −⊠.)

Example (The exceptional flocks [Baker and Ebert (1987) (orders 11,23), Bader (1988), Bonisoli (1988), Johnson (1989), see also Durante (1992)]) There are 3 exceptional flocks of the hyperbolic quadric of PG(3, q), one each for q = 11, 23 and 59 associated with the irregular nearfields of Dickson (1905) of orders 11^{2}, 23^{2} and 59^{2}.

Theorem 3.3 (Bader and Lunardon (1989), Thas (1991)). Any flock of a hyperbolic quadric of PG(3, q), q odd, is linear, Thas or exceptional.

A different approach to these hyperbolic flocks has been given by Bader, Durante, Lunardon, Penttila and the present author ([7]), where they are shown to be the smallest member in each of three infinite families of partial flocks of sizes 12,24 and 60 of the hyperbolic quadric of PG(3, q), for q ≡ −1 modulo 12, 24, 60 respectively, by the use of root systems of the Euclidean space ℝ^{4}.

Thus the problem of flocks of nondegenerate quadrics of PG(3, q) is completely solved. But in the wider setting of circle planes, we also have the problem of flocks of Miquelian Laguerre planes, i.e., flocks of quadratic cones of PG(3, q). There exists a number of different examples of flocks of quadratic cones, and although much of the general theory holds for all q, the situation is somewhat different for q odd and for q even. The problem of classification of these flocks is still very much open for both cases, and one of the aims of this thesis is to contribute towards the case of q odd by constructing new examples. We mention here that the problem of flocks of circle planes arising from oval cones for q even is also still open. When the oval is a translation oval, see Cherowitzo (1998) for the study of these so called α-flocks. Flocks of other oval cones are yet to be considered.

3.1 Flocks of quadratic cones

A flock of a quadratic cone K of PG(3, q) is a partition of the points of K minus the vertex V into pairwise disjoint conics. We may talk of the conics of the flock, as well as the planes of the flock, namely, those planes of PG(3, q) containing the conics of the flock. There are necessarily q conics (planes) of the flock. As the conics of a flock are pairwise disjoint, the planes of the flock necessarily meet pairwise in lines exterior to the cone. If all planes of a flock meet in the same external line, then this flock is called linear.

Suppose two flocks F = {C_{1}, …, C_{q}} and F′ = {C′_{1}, …, C′_{q}} of the quadratic cone K have s > \frac{q−1}{2} conics in common. Without loss of generality, say C_{i} = C′_{i} for i = 1, …, s. If s < q, then there exists a conic C′_{j} of F′ such that C′_{j} ∉ F. Let π′_{j} be the plane of C′_{j} and π_{i} be the plane of C_{i} for i = s + 1, …, q. Now π′_{j} meets π_{i} in a line, and so C′_{j} meets C_{i} in at most two points, for i = s + 1, …, q. Since the conics of F are pairwise disjoint, the size of C′_{j} can be at most 2(q − s), that is, q + 1 ≤ 2(q − s), whence s ≤ \frac{q−1}{2}, a contradiction. Thus s = q, and we see that if two flocks share more than \frac{q−1}{2} conics, they are equal (Thas (1987)).

In Fisher and Thas (1979) there is presented an example of a flock of a quadratic cone for all odd q due to Fisher. The geometric construction is very detailed, so we choose to delay presentation of this example until later. Instead we mention only that the example of Fisher has the property that exactly \frac{q−1}{2} planes of the flock meet in a common line, and that Payne and Thas (1991) [99] have characterised this example as the unique nonlinear flock with this property. (Furthermore, the remaining \frac{q+1}{2} planes meet in a common point, which is not on the common line of the first \frac{q−1}{2} planes.)

We have seen earlier that all quadratic cones of PG(3, q) are equivalent under the action of P Γ L(4, q). Thus we define the quadratic cone K having equation

x_{1}^{2} = x_{0}x_{2},

so that K = {(x_{0}, x_{1}, x_{2}, x_{3})|x_{1}^{2} = x_{0}x_{2}}. The vertex of K is the point V (0, 0, 0, 1).

Any plane of a flock F of K will have an equation of the form ax_{0}+bx_{1}+cx_{2}+x_{3} = 0, and we denote these planes by [a, b, c, 1]^{T}, so that

F = {[a_{t}, b_{t}, c_{t}, 1]^{T} : t ∈ GF (q)}.

If we project from V onto the plane π : x_{3} = 0, then K projects the conic C : x_{1}^{2} = x_{0}x_{2} in π. The intersection of the planes [a_{s}, b_{s}, c_{s}, 1]^{T} and [a_{t}, b_{t}, c_{t}, 1]^{T} for s ≠ t projects onto the line [a_{s} − a_{t}, b_{s} − b_{t}, c_{s} − c_{t}]^{T} of π, which must be external to C.

Thus for F to be a flock of K, the following two equations must have no nontrivial solution (x_{0}, x_{1}, x_{2}) :

(a_{s} − a_{t})x_{0} + (b_{s} − b_{t})x_{1} + (c_{s} − c_{t})x_{2} = 0
x_{1}^{2} = x_{0}x_{2}.

By making the appropriate substitutions, the following equation must have no nontrivial solution X :

(a_{s} − a_{t})^{2}X^{2} + [2(a_{s} − a_{t})(c_{s} − c_{t}) − (b_{s} − b_{t})^{2}]X + (c_{s} − c_{t})^{2} = 0.

Using the quadratic formula when q is odd and using the absolute trace equation when q is even, we arrive at the following important theorem.

Theorem 3.4 (Thas (1987)).

F = {[a_{t}, b_{t}, c_{t}, 1]^{T} : t ∈ GF (q)} is a flock of K

⟺ \begin{cases}(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t}) = ⊠ & ∀s ≠ t (q odd) \\ b_{s} ≠ b_{t} and trace(\frac{(a_{s}+a_{t})(c_{s}+c_{t})}{(b_{s}+b_{t})^{2}}) = 1 & ∀s ≠ t (q even) \end{cases}

The following example of an infinite family of flocks of the quadratic cone was first given by Walker (1976) in the context of translation planes and then by Fisher and Thas (1979) explicitly as flocks.

Example (The Fisher-Thas/Walker flocks [Walker (1976), Fisher and Thas (1979)]) With notation as above, let F = {[t, 3t^{2}, 3t^{3}, 1]^{T} : t ∈ GF (q)} be a set of q planes of PG(3, q). Then F is a flock of the quadratic cone K if and only if q ≡ −1 (mod 3). We prove this using the above theorem.

For q odd,

(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t})
= (3s^{2} − 3t^{2})^{2} − 4(s − t)(3s^{3} − 3t^{3})
= 9(s − t)^{2}(s + t)^{2} − 12(s − t)^{2}(s^{2} + st + t^{2})
= (s − t)^{2}(−3s^{2} + 6st − 3t^{2})
= −3(s − t)^{4}

Now GF (q) has a primitive cube root of unity if and only if 3|q − 1, so q ≡ −1 (mod 3) if and only if there is no cube root of unity in GF (q), if and only if x^{2} + x + 1 is irreducible over GF (q), if and only if Δ = 1 − 4 = −3 is a non-square in GF (q). Hence (b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t}) is a nonsquare for s ≠ t if and only if q ≡ −1 (mod 3).

For q even,

trace(\frac{(a_{s}+a_{t})(c_{s}+c_{t})}{(b_{s}+b_{t})^{2}})
= trace(\frac{(s+t)(s^{3}+t^{3})}{(s^{2}+t^{2})^{2}})
= trace(1 + \frac{s^{3}t+st^{3}}{s^{4}+t^{4}} )
= trace(1 + \frac{(st+s^{2})}{(s^{2}+t^{2})} + (\frac{st+s^{2}}{s^{2}+t^{2}} )^{2})

Now trace(x + x^{2}) = 0 for all x ∈ GF (q), q = 2^{h}, and trace(1) = 1 if and only if h is odd, if and only if q ≡ −1 (mod 3). Note that s^{2} + t^{2} ≠ 0 for s ≠ t as x ↦ x^{2} is one-to-one.

This flock is linear if and only if q = 2.

For the rest of this thesis we will only be concerned with flocks of the quadratic cone, and so we may often call them simply flocks, and drop the reference to the quadratic cone. A central aim of this thesis is to discover new examples of flocks of the quadratic cone. Thus we need to know just what ‘new’ means, that is, we must have a concept of equivalence of flocks. Being partitions of the quadratic cone of PG(3, q), flocks are substructures of PG(3, q), and so naturally we consider equivalence under P Γ L(4, q). Thus two flocks F_{1} and F_{2} of the quadratic cone K of PG(3, q) are isomorphic if and only if there exists an element of P Γ L(4, q) which fixes K and maps F_{1} onto F_{2}. We will discuss the question of isomorphism in more detail later.

44  Chapter 3. Flocks of circle planes

CHAPTER 4 Associated geometries

The study of flocks of the quadratic cone in PG(3, q) is a fascinatingly rich area of research. One of the most appealing aspects is the wide diversity of geometric structures connected with flocks. In this chapter we investigate some of these connections. For recent surveys of this wide area see Payne’s survey [96] in [42] and Johnson and Payne (1997).

4.1 Translation planes arising from linespreads

We saw earlier that translation planes of order q^{t+1} are intimately connected with t-spreads of PG(2t + 1, q). In the following sections we will be interested in the case t = 1, so we now present this result in the geometric manner of Bruck and Bose (1964).

A 1-spread of PG(3, q) is a partition of the points into lines, and so is referred to as a linespread. Equivalently, it is a set of q^{2} + 1 lines, no 2 intersecting. From now on we will only be concerned with linespreads of PG(3, q), and we may refer to them simply as spreads.

Embed PG(3, q) as a hyperplane of PG(4, q), and let S be a linespread of PG(3, q). Define an incidence structure A(S) with points the points of PG(4, q) not in PG(3, q), lines the planes of PG(4, q) meeting PG(3, q) only in a line of S, and with incidence inherited from PG(4, q).

Theorem 4.1 (Bruck and Bose (1964)). A(S) is a translation affine plane of order q^{2}.

Each line l of S corresponds to a class of parallel lines of A(S), so adjoining each such line of S to A(S) as a ‘point at infinity’, and adjoining S to A(S) as a ‘line at infinity’ defines a projective plane P(S), which is a translation plane of order q^{2}.

Let S be a linespread of PG(3, q). If, for any three distinct lines l_{1}, l_{2}, l_{3} of S, all lines of the unique regulus through l_{1}, l_{2}, l_{3} are contained in S, then S is called a regular linespread. For q = 2, the regulus through any three lines contains only the three lines, that is, every linespread of PG(3, 2) is regular.

Theorem 4.2 (Bruck and Bose (1966)). With respect to the André/Bruck-Bose constructions, a linespread of PG(3, q) is regular if and only if the corresponding translation plane is Desarguesian.

We say two linespreads S_{1} and S_{2} of PG(3, q) are isomorphic if and only if there exists an element of P Γ L(4, q) which maps S_{1} onto S_{2} linewise; and so, with respect to the André/Bruck-Bose constructions, two linespreads of PG(3, q) are isomorphic if and only if the corresponding translation planes are isomorphic.

We remark however that not all translation planes of order q^{2} arise from linespreads of PG(3, q), if q is not prime. Consider the case for translation planes of order 16 = 4^{2} = 2^{4}. These can arise from linespreads of PG(3, 4) (t = 1), but also from solidspreads of PG(7, 2) (t = 3). We wish to consider only the case of translation planes arising from linespreads of PG(3, q). In the finite case, the kernel of a translation plane is a field, and so if the largest subfield of the kernel is GF (q), the kernel itself will be GF (q). Thus if we stipulate that the largest subfield of the kernel is GF (q), the translation plane arising from a linespread of PG(3, q) is of order q^{2} with kernel GF (q), in other words, it is of dimension 2 over its kernel. These are precisely the translation planes we will consider in this thesis. Note that the translation planes arising from solidspreads of PG(7, 2) would then be of order 16, but of dimension 4 over their kernel.

Let F = GF (q), where q > 3 and q = p^{h} for some prime p ≠ 3. A function L : F → F is likeable if it satisfies the following conditions :

(i) L(x + y) = L(x) + L(y) for all x, y ∈ F (i.e., L is additive); and

(ii) if y^{2} = x^{2}y − \frac{1}{3}x^{4} + xL(x) then x = y = 0.

This concept and the following ideas were introduced by Kantor (1982) [68]. Consider f(x, y) = xy − \frac{1}{3}x^{3} + L(x), where L is a likeable function on GF (q). Then

G(L) = \{M(x, y) = (\begin{matrix}1 & x & y & f(x, y) \\ 0 & 1 & x & y \\ 0 & 0 & 1 & x \\ 0 & 0 & 0 & 1 \end{matrix}) : x, y ∈ GF (q)\}

is an abelian group of order q^{2}. Define the following set of q^{2} + 1 lines of PG(3, q) :

S(L) = {l_{M} = ⟨(x_{0}, x_{1}, 0, 0)M⟩ : M ∈ G(L)} ∪ {l_{∞} = ⟨(0, 0, x_{2}, x_{3})⟩}.

Clearly l_{∞} does not intersect any other line of S(L), and since G(L) is abelian, to check if two distinct lines l_{M}, l_{N} of S(L)\l_{∞} are skew, we can take M = M(0, 0), the identity element of G(L), and N = N(t, u) where at least one of t, u is nonzero. Then we must have that the following two equations :

x_{0}u + x_{1}t = 0 x_{0}f(t, u) + x_{1}u = 0

have only the trivial solution x_{0} = x_{1} = 0, and hence that u^{2} −tf(t, u) ≠ 0. But this follows directly from the definition of a likeable function, and so S(L) is a linespread of PG(3, q). In fact, S(L) contains the q reguli R_{x} = {⟨(x_{0}, x_{1}, 0, 0)M(x, y)⟩ : y ∈ F }∪{⟨(0, 0, x_{2}, x_{3})⟩}, that is, contains q reguli sharing a common line. The likeable translation plane arising from S(L) via the André/Bruck-Bose constructions is non-Desarguesian.

Now set a = x and z = y/a^{2} in the definition of likeability. Then we have the following lemma determining whether a given additive function is likeable.

Lemma 4.1 (Kantor (1982) [68]). An additive function L : F → F is likeable if and only if the equation

z^{2} − z + \frac{1}{3} − \frac{L(a)}{a^{3}} = 0

has no solution for z ∈ F and a ∈ F \{0}. In particular, if q is odd then L is likeable if and only if L(a)/a^{3} − 1/12 is a nonsquare for all a ∈ F \{0}.

Example (The Kantor likeable translation planes) [Kantor (1982) [68]] Let q = 5^{h}, h > 1 and fix a nonsquare n of GF (q). Define the following additive map from GF (q) to GF (q) :

L(t) = nt^{5} + n^{−1}t.

Then

\frac{L(a)}{a^{3}} − \frac{1}{12}
= na^{2} + \frac{1}{na^{2}} − \frac{1}{2}
= \frac{2n^{2}a^{4}+2−na^{2}}{2na^{2}}
= \frac{n^{2}a^{4}+2na^{2}+1}{na^{2}}
= \frac{(na^{2}+1)^{2}}{na^{2}}

and since −1 is a square, a^{2} ≠ −n^{−1}. Hence L is a likeable function and will define a linespread of PG(3, q) as above. The non-Desarguesian translation planes arising from L are called the Kantor likeable translation planes.

We may represent linespreads of PG(3, q) algebraically as follows. Let M be an arbitrary 4 × 2 matrix over GF (q) of column rank 2. Then the column space ⟨M⟩ of M defines a line l_{M} of PG(3, q). Let l_{M}, l_{N} be two such lines of PG(3, q). Then l_{M} and l_{N} are skew if and only if together they span all of PG(3, q), that is, if and only if the 4 × 4 matrix (M, N) has column rank 4. Let I be the 2 × 2 identity matrix over GF (q), and let A, B be arbitrary 2 × 2 matrices over GF (q). Then the lines

l_{A} = ⟨(\begin{matrix}I \\ A \end{matrix})⟩, l_{B} = ⟨(\begin{matrix}I \\ B \end{matrix})⟩ are skew if and only if det(\begin{matrix}I & I \\ A & B \end{matrix}) ≠ 0, if and only if

det(A − B) ≠ 0.

A spread set is a family S of q^{2} 2 × 2 matrices over GF (q) such that A − B is nonsingular, that is, det(A − B) ≠ 0, ∀A ≠ B ∈ S. Clearly, given a spread set S, if we let

S(S) = \{⟨(\begin{matrix}I \\ A \end{matrix})⟩|A ∈ S\} ∪ \{l_{∞} = ⟨(\begin{matrix}0 \\ I \end{matrix})⟩\},

then S(S) is a linespread of PG(3, q). Any linespread of PG(3, q) is equivalent to a linespread of the form S(S), for some spread set S. Furthermore, every spread set S can be normalised so that

S = \{(\begin{matrix}g(x, y) & h(x, y) \\ x & y \end{matrix}) |x, y ∈ GF (q)\}

for some functions g, h : GF (q)^{2} → GF (q).

Theorem 4.3 (Hiramine, Matsumoto and Oyama (1987), Johnson (1988)). Let

S = \{(\begin{matrix}g(x, y) & h(x, y) \\ x & y \end{matrix}) |x, y ∈ GF (q)\}

be a spread set of 2 × 2 matrices over GF (q). For ω ∈ GF (q)^{2}\GF (q), and \overline{x} = x^{q} ∀x ∈ GF (q^{2}),

\{A_{x,y,x′,y′} = (\begin{matrix}x + ωy & g(x′, y′) + \overline{ω}h(x′, y′) \\ x′ + ωy′ & \overline{x} + \overline{ω}\overline{y} \end{matrix}) |x, y, x′, y′ ∈ GF (q)\}

is a spread set over GF (q^{2}).

This remarkable construction was first given by Hiramine, Matsumoto and Oyama (1987) for characteristic not 2, and then generalised by Johnson (1988). Hence we have the following corollary.

Corollary 4.1. Every linespread of PG(3, q) gives rise to a linespread of PG(3, q^{2}).

In fact, different linespreads of PG(3, q^{2}) can arise from the same equivalence class of linespreads of PG(3, q), depending upon both the choice of the spread set S and the choice of ω. Clearly the process can be iterated, leading to linespreads in PG(3, q^{4}), PG(3, q^{8}), ...., and so there are no sporadic linespreads of PG(3, q) : each linespread of PG(3, q) lies in many infinite families of linespreads. (See page 65 for a brief discussion of sporadicity.)

4.1.1 Flocks and translation planes Denote by Q^{+} the Klein quadric of PG(5, q). An ovoid of Q^{+} is a set of q^{2} + 1 points such that no two of the points are collinear in a line of the quadric. Under the Klein correspondence, such a set of points will correspond to a set of q^{2} + 1 lines of PG(3, q), no two of which are concurrent; that is, an ovoid of the Klein quadric corresponds to a linespread of PG(3, q).

Let F = {π_{1}, …, π_{q}} be a flock of the quadratic cone K of PG(3, q), and embed K as a section of Q^{+} by the solid Σ. Let ⊥ denote the polarity arising from Q^{+}. Then π_{i}^{⊥} is a plane of the same type as π_{i}, and so intersects Q^{+} in a conic C_{i}^{⊥}. Define Ω to be the union of these conics, so Ω = ∪{C_{i}^{⊥} : 1 ≤ i ≤ q}. The polar line l to Σ is tangent to Q^{+} at the vertex V of K, and so each π_{i}^{⊥} will contain l and hence contain V. Since the planes π_{i} of F are distinct, the planes π_{i}^{⊥} will pairwise intersect exactly in l, and so |Ω| = q^{2} + 1.

Now consider two distinct planes π_{i}^{⊥} and π_{j}^{⊥}. Then π_{i} ∩ π_{j} is a line l_{ij} exterior to Q^{+}, as π_{i}, π_{j} are planes of the flock F, and so l_{ij} is an elliptic subspace O^{−}(2, q) of the hyperbolic space O^{+}(6, q) underlying Q^{+}. Recall that the orthogonal direct sum of two elliptic spaces is hyperbolic (“minus times minus is plus”). Thus we have that l_{ij}^{⊥} = ⟨π_{i}^{⊥}, π_{j}^{⊥}⟩ is an elliptic subspace O^{−}(4, q), and so ⟨π_{i}^{⊥}, π_{j}^{⊥}⟩ intersects Q^{+} in an elliptic quadric. Hence no two points of ⟨π_{i}^{⊥}, π_{j}^{⊥}⟩ ∩ Q^{+} are collinear in a line of Q^{+}, and so no two points of Ω are collinear in a line of Q^{+}. Thus we have proven the following result due independently to Walker (1976) and Thas (unpublished, c. 1976).

Theorem 4.4 (Walker (1976), Thas). Ω is an ovoid of the Klein quadric of PG(5, q).

We will see later exactly which types of linespreads may be constructed in this way from flocks of the quadratic cone. In a similar manner, an ovoid of the Klein quadric may also be constructed from any flock of an elliptic quadric or hyperbolic quadric of PG(3, q). With respect to the André/Bruck-Bose constructions and the Thas/Walker construction, two flocks of the quadratic cone are isomorphic if and only if the corresponding translation planes are isomorphic. However, due to the constructions of Hiramine, Matsumoto and Oyama and Johnson, a flock gives rise to many infinite families of translation planes.

If F is a linear flock, so that all planes of F intersect in the common line m, it is clear from the above construction that the corresponding ovoid of the Klein quadric will be the elliptic quadric m^{⊥} ∩ Q^{+}. Under the Klein correspondence, this elliptic quadric corresponds to a regular linespread of PG(3, q), which we have seen gives rise to a Desarguesian translation plane via the André/Bruck-Bose constructions.

Example (The Kantor/Knuth semifield ovoids) [Kantor (1982) [67], Knuth (1965)] Let F = GF (q), where q is odd and not prime. Fix a nonsquare n of F, and let σ and τ be automorphisms of F, at least one of which is nontrivial. Suppose the Klein quadric Q^{+} of PG(5, q) is defined by the quadratic form Q : x_{0}x_{5}+x_{1}x_{4}+x_{2}x_{3}. Define Ω to be the following set of points of Q^{+} :

Ω = {⟨(1, a, b, b^{τ}, −na^{σ}, −b^{τ+1} + na^{σ+1})⟩ : a, b ∈ F } ∪ {⟨(0, 0, 0, 0, 0, 1)⟩}.

Clearly, the point ⟨(0, 0, 0, 0, 0, 1)⟩ is not orthogonal to any other points in Ω. Let

⟨u⟩ = ⟨(1, a, b, b^{τ}, −na^{σ}, −b^{τ+1}+na^{σ+1})⟩ and ⟨v⟩ = ⟨(1, c, d, d^{τ}, −nc^{σ}, −d^{τ+1}+nc^{σ+1})⟩

be two distinct points of Ω\{⟨(0, 0, 0, 0, 0, 1)⟩}. Then with respect to the polar form f of Q we have :

f(u, v) = u_{0}v_{5} + u_{5}v_{0} + u_{1}v_{4} + u_{4}v_{1} + u_{2}v_{3} + u_{3}v_{2}
= −d^{τ+1} + nc^{σ+1} − b^{τ+1} + na^{σ+1} − nac^{σ} − nca^{σ} + bd^{τ} + db^{τ}
= −(b^{τ} − d^{τ})(b − d) + n(a^{σ} − c^{σ})(a − c)
= −(b − d)^{τ+1} + n(a − c)^{σ+1}

Suppose ⟨u⟩ and ⟨v⟩ are collinear in a line of Q^{+}, that is, they are orthogonal. Then f(u, v) = 0 and so

n = \frac{(b − d)^{τ+1}}{(a − c)^{σ+1}} = (\frac{(b − d)^{\frac{τ+1}{2}}}{(a − c)^{\frac{σ+1}{2}}} )^{2}

4.2. Elation generalised quadrangles arising from q-clans  51

since σ and τ are odd. But n is a nonsquare of F, and hence Ω is a set of q^{2} + 1 pairwise noncollinear points of Q^{+}, that is, an ovoid of Q^{+}, given by Kantor (1982) [67]. The translation plane arising from Ω via the Klein correspondence and the André/Bruck-Bose constructions has as its coordinatising ternary field a semifield given by Knuth (1965). If one of σ and τ is the identity automorphism, then Ω lies within some hyperplane H of PG(5, q), and Ω is then an ovoid of the parabolic quadric Q(4, q) defined by H ∩ Q^{+}. In this case, the associated semifield reduces to a commutative semifield first given by Dickson (1905) (see Dickson (1935)). For q = 9, it is equivalent to the commutative semifield of Ganley given in Section 2.1.1.

Now although a flock gives rise to an ovoid of the Klein quadric via the Thas/Walker construction, not every ovoid of the Klein quadric must arise from a flock. In the case that an ovoid of the Klein quadric lies in a hyperplane section of this quadric, so that it is an ovoid of Q(4, q), we call it a translation ovoid if the translation plane arising via the Klein correspondence and André/Bruck-Bose constructions is a semifield plane. A flock giving rise to a semifield translation plane via the Thas/Walker construction is called a semifield flock, and every translation ovoid of Q(4, q) defines a semifield flock. (See Lunardon (1997) and Thas (1997) [116].) Hence the Kantor/Knuth translation ovoid above gives rise to a semifield flock. Furthermore, the translation plane coordinatised by the Ganley commutative semifield (page 31) also arises from a flock, with the associated translation ovoid first being studied by Thas and Payne (1994).

4.2 Elation generalised quadrangles arising from q-clans

We saw earlier that elation generalised quadrangles of order (s, t), s > 1, t > 1, can be constructed from any finite group of order s^{2}t which admits a Kantor (4gonal) family of subgroups. Motivated by this construction technique, Payne (1980) described an appropriate group in terms of a set of q 2 × 2 matrices, later called a q-clan, which could be used to construct an EGQ of order (q^{2}, q). Refining the calculations required in the odd characteristic case, Kantor (1986) used this group to describe two new infinite families of generalised quadrangles. Payne (1985) gave the corresponding simplifications in the even characteristic case and was able to describe a further infinite family, analogous to one of the examples found by Kantor. The work of Thas (1987) then connected this theory to the study of flocks of the quadratic cone.

Let A be a 2 × 2-matrix over GF (q). Then A is anisotropic provided xAx^{T} = 0 if and only if x = (0, 0) ∈ GF (q)^{2}. So an anisotropic matrix represents a nondegenerate binary quadratic form.

Let A = (\begin{matrix}a_{1} & a_{2} \\ a_{3} & a_{4} \end{matrix}) and B = (\begin{matrix}b_{1} & b_{2} \\ b_{3} & b_{4} \end{matrix}). We call A and B equivalent, written

A ≡ B, if a_{1} = b_{1}, a_{4} = b_{4} and a_{2} + a_{3} = b_{2} + b_{3}. Then xAx^{T} = xBx^{T} for all x ∈ GF (q)^{2} if and only if A ≡ B, that is, A and B represent the same quadratic form.

For q odd, (\begin{matrix}a_{1} & a_{2} \\ a_{3} & a_{4} \end{matrix}) ≡ (\begin{matrix}a_{1} & \frac{a_{2}+a_{3}}{2} \\ \frac{a_{2}+a_{3}}{2} & a_{4} \end{matrix} ), and for q even, (\begin{matrix}a_{1} & a_{2} \\ a_{3} & a_{4} \end{matrix}) ≡ (\begin{matrix}a_{1} & a_{2} + a_{3} \\ 0 & a_{4} \end{matrix}),

so in general we often deal with upper triangular matrices, but may use symmetric matrices if q is odd.

Let C = \{A_{t} ≡ (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\} be a set of q 2 × 2-matrices over GF (q),

indexed by the elements of GF (q). Then C is called a q-clan if and only if the pairwise difference of any two distinct matrices in C is anisotropic. Clearly the difference of two equivalent matrices cannot be anisotropic, and so a q-clan is a set of q pairwise inequivalent matrices, i.e., a set of q equivalence class representatives. If, for three functions a : t ↦ a_{t}, b : t ↦ b_{t}, c : t ↦ c_{t} from GF (q) to GF (q),

C = \{A_{t} ≡ (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\} is a q-clan, then it is clear that the functions a

and c must be one-to-one, and if q is even, then b must also be one-to-one.

Let A_{s} = (\begin{matrix}a_{s} & b_{s} \\ 0 & c_{s} \end{matrix}) and A_{t} = (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}). Solving x(A_{s} − A_{t})x^{T} = 0 for x ∈ GF (q)^{2}

leads to the following useful condition on a set of matrices being a q-clan, where the determinant notation for q odd is used for ease of reference :

Theorem 4.5. Let a : t ↦ a_{t}, b : t ↦ b_{t}, c : t ↦ c_{t} be three functions from GF (q)

to GF (q). Then C = \{A_{t} ≡ (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\} is a q-clan if and only if the

following holds :

(i) −det((A_{s} + A_{s}^{T}) − (A_{t} + A_{t}^{T})) = (b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t}) is a nonsquare in GF (q) for distinct s, t ∈ GF (q), if q is odd;

(ii) b_{s} ≠ b_{t} and trace(\frac{(a_{s}+a_{t})(c_{s}+c_{t})}{(b_{s}+b_{t})^{2}}) = 1 for distinct s, t ∈ GF (q), if q is even.

The functions a(t) = t, b(t) = bt and c(t) = ct for x^{2} + bx + c irreducible over GF (q) define a q-clan, called a classical q-clan.

Example (The Kantor/Payne monomial q-clans) [Kantor (1986), Payne

(1985)] Let C = \{A_{t} ≡ (\begin{matrix}t & 5t^{3} \\ 0 & 5t^{5} \end{matrix}) : t ∈ GF (q)\} be a set of q 2 × 2 matrices over

GF (q). Then C is a q-clan if and only if q ≡ ±2 (mod 5). We prove this using the above theorem.

For q odd,

(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t})
= (5s^{3} − 5t^{3})^{2} − 4(s − t)(5s^{5} − 5t^{5})
= 25(s − t)^{2}(s^{2} + st + t^{2})^{2} − 20(s − t)^{2}(s^{4} + s^{3}t + s^{2}t^{2} + st^{3} + t^{4})
= (s − t)^{2}(5s^{4} + 30s^{3}t + 55s^{2}t^{2} + 30st^{3} + 5t^{4})
= 5(s − t)^{2}(s^{2} + 3st + t^{2})^{2}

Now, for q odd, 5 is a non-square in GF (q) if and only if q ≡ ±2 (mod 5). So s^{2} + 3st + t^{2} ≠ 0 for s ≠ t, and hence (b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t}) is a nonsquare for s ≠ t if and only if q ≡ ±2 (mod 5).

For q even,

trace(\frac{(a_{s}+a_{t})(c_{s}+c_{t})}{(b_{s}+b_{t})^{2}})
= trace(\frac{(s+t)(s^{5}+t^{5})}{(s^{3}+t^{3})^{2}})
= trace(1 + \frac{s^{5}t+st^{5}}{s^{6}+t^{6}} )
= trace(1 + \frac{s^{2}t+st^{2}}{s^{3}+t^{3}} + (\frac{s^{2}t+st^{2}}{s^{3}+t^{3}} )^{2})
= trace(1)

Now, for q even, trace(1)=1 if and only if q = 2^{h}, h odd if and only if q ≡ ±2 (mod 5). Note that s^{3} + t^{3} ≠ 0 for s ≠ t as x ↦ x^{3} is one-to-one, since h is odd implies 3 is coprime to 2^{h} − 1. Hence trace(\frac{(a_{s}+a_{t})(c_{s}+c_{t})}{(b_{s}+b_{t})^{2}}) = 1 for s ≠ t if and only if q ≡ ±2 (mod 5).

This q-clan is classical if and only if q = 2 or q = 3. It was first given by Kantor (1986) for q odd, and then by Payne (1985) for q even.

An appropriate group

Let F = GF (q) and let K = {(α, c, β) ∈ F^{2} × F × F^{2} : α, β ∈ F^{2}, c ∈ F }. Define a binary operation on K by

(α, c, β) ∘ (α′, c′, β′) = (α + α′, c + c′ + β(α′)^{T} , β + β′).

Then K is a group of order q^{5} with (α, c, β)^{−1} = (−α, α · β − c, −β). The centre of K is Z = Z(K) = {(0, c, 0) : c ∈ F }.

Let C be a set of q 2 × 2 matrices over F, indexed by the elements of F. Suppose

A_{0} = (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) and put K_{t} = A_{t} + A_{t}^{T} for all A_{t} ∈ C. Define the following subgroups of K :

A(∞) = {((0, 0), 0, β) ∈ K : β ∈ F^{2}};
A(t) = {(α, αA_{t}α^{T} , αK_{t}) ∈ K : α ∈ F^{2}} ∀t ∈ F;
A^{∗}(∞) = A(∞)Z = {((0, 0), c, β) ∈ K : c ∈ F, β ∈ F^{2}};
A^{∗}(t) = A(t)Z = {(α, c, αK_{t}) ∈ K : α ∈ F^{2}, c ∈ F } ∀t ∈ F.

Put F̃ = F ∪ {∞}, J (C) = {A(t) : t ∈ F̃}, J^{∗}(C) = {A^{∗}(t) : t ∈ F̃}. Then Payne (1980) proved the following.

Theorem 4.6 (Payne (1980)). The triple (K, J (C), J^{∗}(C)) satisfies the conditions K1 and K2, and so forms a Kantor (4-gonal) family for K if and only if C is a q-clan. The resulting GQ, written GQ(C), is an EGQ of order (q^{2}, q).

These are precisely the EGQs we will consider in this thesis, namely EGQs of order (q^{2}, q) arising from a q-clan. From Higman’s inequality (2.3) we see that GQs of order (s^{2}, s) are extremal. In fact, all known GQs of order (s^{2}, s) are EGQs, and except for the dual of T_{3}(Ω), they all arise from the elation group P SU(4, q^{2}) of the classical GQ H(3, q^{2}). This elation group G = P SU(4, q^{2}) is a special group of order q^{5}, that is, its centre Z(G), which has order q, is equal to both its derived subgroup G′ and to its Frattini subgroup Φ(G) : Z(G) = G′ = Φ(G).

4.2.1 Flocks, translation planes and elation generalised quadrangles We have seen a similar condition to that for q-clans before. Indeed, Thas (1987) gave the connection between q-clans and flocks of the quadratic cone.

Let K be the quadratic cone of PG(3, q) defined earlier, that is, with equation Q : x_{1}^{2} = x_{0}x_{2}. Then we have the following :

Theorem 4.7 (Thas (1987)). For three functions a : t ↦ a_{t}, b : t ↦ b_{t}, c : t ↦ c_{t} from GF (q) to GF (q),

C = \{A_{t} ≡ (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\} is a q-clan

⟺ F(C) = {[a_{t}, b_{t}, c_{t}, 1]^{T} : t ∈ GF (q)} is a flock of K

Consequently, we may construct the spread sets of the associated Thas/Walker

4.3. Fundamental Theorem of q-clan geometry  55

linespreads directly from the q-clan. Let C = \{A_{t} = (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\} be a q-clan, and define the following set of q^{2} 2 × 2 matrices over GF (q) :

S(C) = \{B_{s,t} = A_{t} + (\begin{matrix}0 & s \\ −s & 0 \end{matrix}) |s, t ∈ GF (q)\}.

Then

det(B_{s,t} − B_{s′,t′}) = (s − s′)^{2} + (b_{t} − b_{t′})(s − s′) + (a_{t} − a_{t′})(c_{t} − c_{t′}) = 0

⇔ \begin{cases}(b_{t} − b_{t′})^{2} − 4(a_{t} − a_{t′})(c_{t} − c_{t′}) = □ & (q odd) \\ b_{t} + b_{t′} = 0 or trace(\frac{(a_{t}+a_{t′})(c_{t}+c_{t′})}{(b_{t}+b_{t′})^{2}}) = 0 & (q even). \end{cases}

Since C is a q-clan, if B_{s,t} − B_{s′,t′} is singular, this implies t = t′, and so (s − s′)^{2} = 0, giving s = s′. Hence S(C) is a spread set, and gives rise to a linespread S(C) of PG(3, q).

That the spread set arising directly from a q-clan is equivalent to the spread set arising from the corresponding flock by the Thas/Walker construction was shown using Plücker coordinates by Gevaert and Johnson (1988).

Note that for two 2 × 2 matrices A, B, A ≡ B if and only if B = A + s (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix})

for some s ∈ GF (q). So a spread set arising from a q-clan consists of q equivalence classes each consisting of q equivalent matrices. Just as the associated ovoid of the Klein quadric is the union of q conics pairwise intersecting in a common point, this natural partition of a spread set arising from a q-clan hints at the following characterisation of the linespreads arising from flocks.

Theorem 4.8 (Gevaert, Johnson and Thas (1988)). A flock of a quadratic cone in PG(3, q) is equivalent to a linespread S of PG(3, q), where S is the union of q reguli which pairwise intersect in the same line.

Such a linespread is said to admit a regular conical cover. For example, we have seen earlier that the linespreads arising from the Kantor likeable functions have exactly this property, and so there arise flocks of the quadratic cone, called Kantor likeable flocks.

4.3 Fundamental Theorem of q-clan geometry

We have seen in the previous sections that the studies of flocks of a quadratic cone of PG(3, q), of elation generalised quadrangles of order (q^{2}, q) arising from qclans, and of translation planes of order q^{2} and dimension two over their kernel are intimately connected. These connections give rise to the question of isomorphism of and between these “q-clan geometries”. To assist in answering these questions, the so-called Fundamental Theorem of q-clan geometry was developed by a number of authors, the main contributor being Payne (see Payne and Rogers (1990), Bader, Lunardon and Payne (1994), Payne (1996), Johnson and Payne (1997)). Here we present a complete statement of the theorem in context with the many associated structures we have seen. In the following chapter we will apply this theorem to the calculation of the automorphism groups of some known examples.

Let C = \{A_{t} ≡ (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\} be a q-clan. If 0 ≠ μ ∈ GF (q), σ ∈

AutGF (q), B = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL(2, q), and if x, y, z are any fixed elements of GF (q),

then replacing A_{t} ∈ C with A′_{t} ≡ μBA_{t}^{σ}B^{T} + (\begin{matrix}x & y \\ 0 & z \end{matrix}) gives a new q-clan C′ = \{A′_{t} :

t ∈ GF (q)}. Hence if C = {A_{t} : t ∈ GF (q)} and C′ = {A′_{t} : t ∈ GF (q)} are q-clans, we say C and C′ are equivalent, written C ∼ C′, provided there exist the following : 0 ≠ μ ∈ GF (q), B ∈ GL(2, q), σ ∈ AutGF (q), M a 2 × 2-matrix over GF (q), and a permutation π : t ↦ \overline{t} on GF (q) for which the following holds :

A′ ; \overline{t} ≡ μBA_{t}^{σ}B^{T} + M, ∀t ∈ GF (q).

Now let C = \{A_{t} ≡ (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\} be a q-clan normalised so that A_{0} =

(\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}). Furthermore, assume without loss of generality that when q is odd the

members of C are indexed so that a_{t} = t for all t ∈ GF (q), and when q is even that b_{t} = t for all t ∈ GF (q). Let C′ be a second q-clan normalised in the same manner.

The Fundamental Theorem of q-clan geometry shows how to interpret each equivalence between C and C′ as an explicit isomorphism between the corresponding geometries : flocks of the quadratic cone, elation generalised quadrangles of order (q^{2}, q), or linespreads of PG(3, q).

Theorem 4.9 (Fundamental theorem of q-clan geometry). For any prime

power q, let C = \{A_{t} ≡ (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\} and C′ = \{A′_{t} ≡ (\begin{matrix}a′_{t} & b′_{t} \\ 0 & c′_{t} \end{matrix}) : t ∈

GF (q)} be two (not necessarily distinct) normalised q-clans. Then the following are equivalent :

(i) C ∼ C′;

(ii) The generalised quadrangles GQ(C) and GQ(C′) are isomorphic by an isomorphism mapping (∞) to (∞), [A(∞)] to [A′(∞)], and (\overline{0}, 0, \overline{0}) to (\overline{0}, 0, \overline{0});

(iii) The flocks F(C) and F(C′) are isomorphic;

(iv) The linespreads S(C) and S(C′) are isomorphic.

Moreover, these four conditions hold if and only if there exist 0 ≠ λ ∈ GF (q), σ ∈ AutGF (q), B ∈ GL(2, q) and a permutation π : t ↦ \overline{t} of GF (q) for which

A′ ; \overline{t} ≡ λB^{−1}A_{t}^{σ}B^{−T} + A′ ; \overline{0} ∀t ∈ GF (q).

The corresponding isomorphism θ : GQ(C) ↦ GQ(C′) is given by

θ = θ(λ, B, σ, π) : (α, c, β) ↦ (α^{σ}B, λc^{σ} + α^{σ}BA′ ; \overline{0}B^{T} (α^{σ})^{T} , λβ^{σ}B^{−T} + α^{σ}BK′ ; \overline{0}).

If B^{−1} = (\begin{matrix}a & b \\ c & d \end{matrix}), the corresponding isomorphism T_{θ} : F(C) ↦ F(C′) is defined for

planes of the flocks by

T_{θ} : [\begin{matrix}a_{t} \\ b_{t} \\ c_{t} \\ 1 \end{matrix}] ↦ [\begin{matrix}a′ ; \overline{t} \\ b′ ; \overline{t} \\ c′ ; \overline{t} \\ 1 \end{matrix}] = (\begin{matrix}λa^{2} & λab & λb^{2} & a′ ; \overline{0} \\ 2λac & λ(ad + bc) & 2λbd & b′ ; \overline{0} \\ λc^{2} & λcd & λd^{2} & c′ ; \overline{0} \\ 0 & 0 & 0 & 1 \end{matrix}) [\begin{matrix}a_{t}^{σ} \\ b_{t}^{σ} \\ c_{t}^{σ} \\ 1 \end{matrix}] .

The corresponding isomorphism between the associated translation planes is induced by the following isomorphism T : S(C) ↦ S(C′) of the associated linespreads

T : (x_{0}, x_{1}, x_{2}, x_{3}) ↦ (x_{0}, x_{1}, x_{2}, x_{3}) (\begin{matrix}λ^{−1}B & λ^{−1}BA′ ; \overline{0} \\ 0 & B^{−T} \end{matrix}) .

4.4 Hyperbolic fibrations

In this section we discuss a topic which has very recently been connected with the study of flocks of the quadratic cone.

A hyperbolic fibration of PG(3, q) is a set of q − 1 hyperbolic quadrics and two lines which together partition the points of PG(3, q). The classical example of a hyperbolic fibration comes from a pencil of quadrics, but several other families are known to exist.

An important motivation for the study of hyperbolic fibrations is that they give rise to many linespreads of PG(3, q), and hence to many translation planes. Each choice of one regulus for each quadric in a hyperbolic fibration, together with the lines of the fibration, will define a linespread of PG(3, q), and so 2^{q−1} linespreads arise. Each of these linespreads contains q − 1 disjoint reguli, and is said to admit a regular elliptic cover.

A hyperbolic fibration is called regular if the two lines in the fibration are polar to each other with respect to each of the q − 1 hyperbolic quadrics in the fibration. All known hyperbolic fibrations are regular, and these are the type we are interested in here.

For a regular hyperbolic fibration H of PG(3, q), if we choose bases {v_{0}, v_{1}} and {v_{2}, v_{3}} for the two lines l_{0} and l_{∞} of H, then with respect to the basis {v_{0}, v_{1}, v_{2}, v_{3}} of the vector space underlying PG(3, q) any quadric in H will have an equation of the form ax_{0}^{2} + bx_{0}x_{1} + cx_{1}^{2} + dx_{2}^{2} + ex_{2}x_{3} + fx_{3}^{2}, for some a, b, c, d, e, f ∈ GF (q). We abbreviate such a quadric by V [a, b, c, d, e, f], and call (a, b, c) the front half and (d, e, f) the back half. Since l_{0} = V [0, 0, 0, d, e, f] for any d, e, f ∈ GF (q) with dx_{2}^{2} + ex_{2}x_{3} + fx_{3}^{2} an anisotropic quadratic form, and similarly l_{∞} = V [a, b, c, 0, 0, 0] for any a, b, c ∈ GF (q) with ax_{0}^{2} + bx_{0}x_{1} + cx_{1}^{2} an anisotropic quadratic form, a regular hyperbolic fibration may be written as

H = {V [d, e, f, 0, 0, 0], V [0, 0, 0, d, e, f]} ∪ {V [a_{i}, b_{i}, c_{i}, d, e, f] : i = 1, …, q − 1}

for some fixed d, e, f ∈ GF (q) with dx^{2} + exy + fy^{2} an anisotropic quadratic form. Such a hyperbolic fibration is said to have constant back half, and in a similar manner we may describe a hyperbolic fibration with constant front half.

It can be shown that H is a hyperbolic fibration if and only if a_{i}x_{0}^{2} + b_{i}x_{0}x_{1} + c_{i}x_{1}^{2} is an anisotropic form over GF (q) for all i = 1, …, q − 1 and (a_{i} − a_{j})x_{0}^{2} + (b_{i} − b_{j})x_{0}x_{1} + (c_{i} − c_{j})x_{1}^{2} is an anisotropic form over GF (q) whenever i ≠ j (see Baker, Dover, Ebert and Wantz (1998) for the q odd case). Hence we have the following result.

Theorem 4.10 (Baker, Ebert and Penttila (submitted)). With notation as above, H is a hyperbolic fibration if and only if

C = \{A_{i} ≡ (\begin{matrix}a_{i} & b_{i} \\ 0 & c_{i} \end{matrix}) : i = 1, …, q − 1\} ∪ \{(\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix})\} is a q-clan.

Thus we have a bijection between regular hyperbolic fibrations with constant back half and flocks of a quadratic cone with one plane specified, namely the plane π_{0} cor-

responding to A_{0} = (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}). Furthermore, Baker, Ebert and Penttila (submitted)

prove the following, where equivalence of hyperbolic fibrations is naturally defined under P Γ L(4, q).

Theorem 4.11 (Baker, Ebert and Penttila (submitted)). The number of mutually inequivalent regular hyperbolic fibrations with constant back half arising from a given flock F of the quadratic cone is the number of orbits of the stabiliser of F in P Γ L(4, q) on the planes of F.

Thus each flock of the quadratic cone may give rise to a number of inequivalent examples of regular hyperbolic fibrations. Unfortunately, none of the five infinite families of regular hyperbolic fibrations (for q odd) known before these connections were made gave rise to new examples of flocks.

We have noted that a hyperbolic fibration also gives rise to 2^{q−1} linespreads of PG(3, q) admitting regular elliptic covers. Other than the case of the regular linespread, none of the linespreads arising from a hyperbolic fibration can be isomorphic to the linespread arising from the associated flock via the Thas/Walker construction, as this linespread admits a regular conical cover. Thus with the construction of hyperbolic fibrations from flocks of the quadratic cone, we see that a nonlinear flock gives rise to at least two inequivalent linespreads of PG(3, q), and hence at least two translation planes (before employing the construction of Hiramine, Matsumoto and Oyama, and Johnson). For a discussion concerning the equivalence of the linespreads arising from a hyperbolic fibration, see Baker, Ebert and Penttila (submitted).

60  Chapter 4. Associated geometries

CHAPTER 5 Application of the Fundamental Theorem of q-clan geometry

The automorphism group of a geometric structure describes the symmetries inherent in that structure, and so provides a great deal of information about the nature of the structure. In particular, we can use the automorphism groups of certain structures of the same type to distinguish them from one another, which is extremely important in the context of describing ‘new’ examples of a certain structure. The Fundamental Theorem of q-clan geometry, presented in Section 4.3, shows how to interpret equivalence of q-clans as explicit isomorphisms between any of the associated geometries, and so can be used to describe the automorphism group of a particular object. Indeed, the explicit description of the isomorphisms between the associated flocks provides a concrete way of describing the isomorphisms between the associated q-clans and GQs in terms of semilinear mappings of three dimensional projective space. This enables the description of certain collineations of the GQ arising from a q-clan, from which we can calculate the order of the automorphism group of the GQ (and more). Often the order of the automorphism group, together with the necessary conditions on the values of q, are sufficient to distinguish examples from one another.

In this section we discuss the ideas behind applying the Fundamental Theorem of q-clan geometry to the calculation of the automorphism group of a GQ arising from a q-clan. A complete treatment of the Fundamental Theorem is given by Payne (1996); here we expound these ideas, and demonstrate their use by calculating the automorphism group of a recently discovered semifield example.

We will work in the group coset geometry model of an elation generalised quadrangle arising from a q-clan. So let GQ(C) be the EGQ of order (q^{2}, q) arising from a q-clan C and the special group K of Section 4.2. Suppose C is normalised in the manner of Section 4.3, that is, each matrix A_{t} in C is symmetric or upper triangular according as q is odd or even, A_{0} is the zero matrix, and a_{t} = t if q is odd and b_{t} = t if q is even.

We may assume that GQ(C) is not the classical GQ H(3, q^{2}), since in this case the automorphism group is the classical group P Γ U(4, q^{2}), which is well known from the theory of polar spaces. We now have the following useful result.

Theorem 5.1 (Payne and Thas (1991) [99]). If GQ(C) is not classical, then the point (∞) is fixed by all collineations of GQ(C).

62  Chapter 5. Application of the Fundamental Theorem of q-clan geometry

As an aside, we note that this result is based upon the notion of a GQ having Property (G) at some point. This combinatorial property was introduced by Payne (1989), where it was proved that each GQ arising from a q-clan has Property (G) at its point (∞). Payne and Thas (1991) [99] then showed that if a GQ arising from a q-clan has Property (G) at some other point collinear with (∞), then the GQ is classical. We will not discuss Property (G) in any detail here, but mention that Thas (1999) has characterised the GQs arising from q-clans (flock GQs) in terms of this Property (G).

Now since all collineations of GQ(C) fix the point (∞), any collineation will map elements of GQ(C) to elements of the same type, with respect to distance from (∞) in the incidence graph of GQ(C). Hence to determine the full automorphism group AutGQ(C) of GQ(C), we need only determine the action of AutGQ(C) on one type of element of GQ(C), and the kernel of that action if any exists. We choose to describe the action of AutGQ(C) on the set X of points not collinear with (∞). The reason for this choice is as follows.

A standard result from permutation group theory says that a finite group acting on a set can be written as a semidirect product of some regular normal subgroup by the stabiliser of some point of the set. Now consider the group K acting by right multiplication on the set X of points not collinear with (∞). This induces a subgroup of AutGQ(C) acting regularly on X and fixing every line through (∞), that is, right multiplication by K induces a subgroup of elations of GQ(C). This subgroup has the maximum possible order (s^{2}t = q^{5}) of a subgroup of elations, due to the regular action on the set X of size q^{5}, and so K induces the full elation group of the EGQ GQ(C). Moreover, since GQ(C) is nonclassical, the conjugate of an elation is again an elation, and so the induced elation group is also normal in AutGQ(C). Hence we naturally have a normal subgroup K of AutGQ(C) acting regularly on the set X of points not collinear with (∞), and so using the standard result mentioned above we need only determine the stabiliser in AutGQ(C) of some point of X. The kernel of the action of AutGQ(C) on the set X is clearly trivial.

Let O = ((0, 0), 0, (0, 0)). Then O is a point not collinear with (∞), and so we may determine the stabiliser G of O in AutGQ(C). Note that G is analogous to the translation complement in the theory of translation planes, and so we might call G the elation complement of GQ(C). Similar to above, we now consider the action of G on the set Y of lines through (∞). If G_{[A(∞)]} is the stabiliser of the line [A(∞)] through (∞), then we may write G as the following union of cosets

G = ⋃_{[A(s)]∈O}(G_{[A(∞)]})g_{[A(s)]}

where g_{[A(s)]} is some element of G taking [A(∞)] to the line [A(s)] in the orbit O of [A(∞)] under G. In other words, to determine the action of G on Y we may first determine the stabiliser of the line [A(∞)] in G and then determine for each line [A(s)] through (∞) a collineation in G taking [A(∞)] to [A(s)]. Furthermore, the kernel of the action of G on Y may not be trivial, and so this kernel must also be determined.

Let us first consider the stabiliser H of [A(∞)] in G. We are now asking for all collineations of GQ(C) which fix O and [A(∞)] (and, of course, (∞)). If we take C = C′ in the statement of the Fundamental Theorem of q-clan geometry (Section 4.3), then such collineations exist if and only if there exist certain parameters defining the collineations which satisfy a particular condition. The concrete way to calculate the freedom of these parameters, and hence determine the number and form of these collineations, is to work with the equivalent condition on the planes of the flock F(C) associated with the q-clan C.

Now suppose there exists a collineation θ ∈ G taking the line [A(∞)] to some other line [A(t)] through (∞). As an automorphism of GQ(C), θ will map the 4-gonal family corresponding to C to some other 4-gonal family of K corresponding to the line [A(t)]. In general, we cannot assume that a q-clan for this second 4-gonal family will be clearly present. However, Payne (1996) describes an automorphism i_{s}, called the shift-flip, which replaces the (normalised) q-clan C with a (normalised) q-clan C^{i_{s}} corresponding to the line [A(s)] through (∞) (i_{∞} is the identity automorphism), and uses this to prove the following useful theorem.

Theorem 5.2 (Payne (1996)). Let C be a normalised q-clan. Then for arbitrary s, t ∈ GF (q) ∪ {∞}, there is a collineation of GQ(C) mapping [A(s)] to [A(t)] if and only if C^{i_{s}} is equivalent to C^{i_{t}}.

Before continuing, note that each q-clan C^{i_{s}} gives rise to a flock of the quadratic cone, and so to each line through (∞) there corresponds a flock. Since there may or may not be a collineation of GQ(C) mapping [A(s)] to [A(t)], the corresponding flocks may or may not be isomorphic by the Fundamental Theorem 4.9. Hence there arise q + 1 flocks from GQ(C), which may or may not be pairwise equivalent. This will be clarified further in the following chapter, but will also be important in the following section.

Thus to determine a collineation taking [A(∞)] to the line [A(t)] through (∞), we first compute the q-clan C′ = C^{i_{t}} corresponding to [A(t)]. Then by the above Theorem, there exists a collineation taking [A(∞)] to [A(t)] if and only if C is equivalent to C′, which by the Fundamental Theorem (Section 4.3) is true if and only if there exist certain parameters defining this collineation which satisfy a particular condition. Again, we attempt to calculate the freedom within these parameters by considering the equivalent condition on the planes of the flocks F(C) and F(C′).

Finally, Payne (1996) Proposition 3 shows that the kernel of the action of G on the set of lines through (∞) has order q − 1, except in the case of the Kantor/Knuth semifield example, where the kernel has order 2(q − 1) if σ^{2} ≠ 1 and order 4(q − 1) if σ^{2} = 1 ≠ σ.

5.1 A semifield example

As mentioned earlier, the Fundamental Theorem of q-clan geometry was developed over a number of years by various authors. The ideas grew out of attempts to calculate the automorphism groups of some of the known examples of GQs, with much of the earlier work developing the Fundamental Theorem occurring for characteristic 2. The paper by Payne (1992) collects together many of these calculations and presents a rather complete treatment of the automorphism groups for each of the infinite families known at that time. Here we will demonstrate the use of the Fundamental Theorem by applying it to the calculation of the automorphism group of a recently discovered semifield example.

A flock which gives rise to a translation plane coordinatised by a semifield is called a semifield flock. Johnson (1987) showed that any semifield flock for q even must be linear, and so for this section we consider q to be odd, and consider a nonlinear semifield flock F(C) arising from a q-clan

C = \{A_{t} ≡ (\begin{matrix}t & b_{t} \\ 0 & c_{t} \end{matrix}) : t ∈ GF (q)\}.

Let f be the function t ↦ b_{t} and g be the function t ↦ c_{t}. Then F(C) is a semifield flock if and only if f and g are additive (see Gevaert and Johnson (1988)). The generalised quadrangles arising from semifield flocks have associated with them a number of interesting structures, which have been deeply studied in recent times.

The point-line dual of the GQ arising from a semifield flock is an EGQ for which the elation group is abelian (see Johnson (1987)). Such EGQs are called translation generalised quadrangles (TGQs), and can be constructed from certain partial spreads of a higher dimensional projective space called eggs, which generalise the notion of an ovoid (see Payne and Thas (1984)). Such an egg gives rise to an egg in the dual space, from which a TGQ, called the translation dual, can be constructed as before, however now the translation dual need not be the point-line dual of a GQ arising from a flock (see Payne (1989)). For recent work on these topics see Blokhuis, Lavrauw and Ball (submitted), Lavrauw and Penttila (2001), Thas (1994), Thas (1997) [115], Thas (1999), Thas (2001) [118], K. Thas (2001), K. Thas [124], [125].

As the functions defining semifield flocks are additive, Bader and Lunardon (1994) have proved the following.

Theorem 5.3 (Bader and Lunardon (1994)). If F(C) is a nonlinear semifield flock for which there exists a polynomial h(t) over GF (q) such that for a fixed nonsquare n of GF (q) the equation f^{2}(t) + 4tg(t) = nh^{2}(t) is a polynomial identity, then F(C) arises either from the Kantor/Knuth translation ovoid (page 50) or from the Ganley semifield (page 31).

Hence any further example of a semifield flock cannot belong to an infinite family of semifield flocks. The term sporadic is often used to describe an example of a particular structure which does not belong to an infinite family of that type of structure, and so any further semifield flock must be sporadic (as a semifield flock).

Indeed, there exists such a further example. As a result of a computer search, Penttila and Williams (2000) described a translation ovoid of Q(4, q) for q = 3^{5}, not equivalent with any previously known example. Bader, Lunardon and Pinneri (1999) then described the associated semifield flock, and studied the generalised quadrangles arising, showing them all to be new. In a recent paper, Bader, Ghinelli and Penttila (2001) gave a computerfree construction of this sporadic flock, linking it with the Mathieu group M_{11}. This group is considered to be sporadic as a finite simple group, and so provides further evidence for the sporadicity of this example.

The q-clan associated to the Penttila-Williams/Bader-Lunardon-Pinneri flock has the following form :

C = \{A_{t} ≡ (\begin{matrix}t & −t^{27} \\ −t^{27} & −t^{9} \end{matrix}) : t ∈ GF (3^{5})\}

We will now use the Fundamental Theorem of q-clan geometry to calculate the automorphism group of the GQ, GQ(C), arising from this q-clan C. Firstly, we wish to determine those collineations of GQ(C) fixing the line [A(∞)] and the point O = ((0, 0), 0, (0, 0)). By the Fundamental Theorem 4.9, this is equivalent to determining

all permutations π : t ↦ \overline{t} of GF (q), 0 ≠ λ ∈ GF (q), σ ∈ AutGF (q), (\begin{matrix}a & b \\ c & d \end{matrix}) ∈

GL(2, q) for which

[ \overline{t} ; \overline{t}^{27} ; −\overline{t}^{9} ; 1] = (\begin{matrix}λa^{2} & λab & λb^{2} & \overline{0} \\ 2λac & λ(ad + bc) & 2λbd & \overline{0}^{27} \\ λc^{2} & λcd & λd^{2} & −\overline{0}^{9} \\ 0 & 0 & 0 & 1 \end{matrix}) [\begin{matrix}t^{σ} \\ t^{27σ} \\ −t^{9σ} \\ 1 \end{matrix}]

for all t ∈ GF (q). Thus we obtain three equations :

(i) \overline{t} = λa^{2}t^{σ} + λabt^{27σ} − λb^{2}t^{9σ} + \overline{0}; (ii) \overline{t}^{27} = 2λact^{σ} + λ(ad + bc)t^{27σ} − 2λbdt^{9σ} + \overline{0}^{27}; (iii) −\overline{t}^{9} = λc^{2}t^{σ} + λcdt^{27σ} − λd^{2}t^{9σ} − \overline{0}^{9}. From (i) and (iii) we have :

\begin{matrix}−\overline{t}^{9} & = −λ^{9}a^{18}t^{9σ} − λ^{9}a^{9}b^{9}t^{σ} + λ^{9}b^{18}t^{81σ} − \overline{0}^{9} \\ & = λc^{2}t^{σ} + λcdt^{27σ} − λd^{2}t^{9σ} − \overline{0}^{9}. \end{matrix} (∗)

The coefficient of t^{81σ} must be 0, implying b = 0. The coefficient of t^{27σ} must also

be 0, and since (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL(2, q) and b = 0, we have c = 0. Thus from (i) and

(ii) we now have :

\begin{aligned}\overline{t}^{27} & = λ^{27}a^{54}t^{27σ} + \overline{0}^{27} \\ & = λadt^{27σ} + \overline{0}^{27}, \end{aligned}

which implies λ^{27}a^{54} = λad, and so

d = λ^{26}a^{53}.

From the coefficients of t^{9σ} in (∗) we have λ^{9}a^{18} = λd^{2} = λ^{53}a^{106}, and so

(λa^{2})^{44} = 1.

Hence λ = \frac{w}{a^{2}} where w^{44} = 1. Since 3^{5} − 1 = 242, the 44th roots of unity in GF (3^{5}) are just the 22nd roots of unity, and so w is a 22nd root of unity. Now for q = 3^{5}, choose a in q − 1 ways and let σ ∈ AutGF (q). Put λ = \frac{w}{a^{2}} where w is a 22nd root of unity, and put d = w^{4}a, with b = c = 0. Then the three equations (i), (ii), (iii) above all hold. Hence the subgroup of AutGQ(C) fixing [A(∞)] and O = ((0, 0), 0, (0, 0)) has order

5.22.(q − 1).q = 6468660.

Now following the discussion above, we must determine, for each t ∈ GF (q), a collineation taking [A(∞)] to the line [A(t)] through (∞). However, if we allow ourselves to make a forward reference to the material in the following chapter, we can combine a result there with the following useful theorem to show that for the Penttila-Williams/Bader-Lunardon-Pinneri semifield flock F(C), no such collineations of GQ(C) exist. As briefly indicated earlier (page 64), to each line through (∞) of a GQ arising from a q-clan there corresponds a flock of the quadratic cone. Denote these q + 1 flocks by F_{t}, where [A(t)] is a line through (∞), for t ∈ GF (q) ∪ {∞}.

Theorem 5.4 (Lunardon (1991)). Let F_{∞} be a nonlinear semifield flock, and let F_{1}, …, F_{q} be the q flocks corresponding to the q lines through (∞) of the GQ arising from F_{∞}, other than the line [A(∞)] corresponding to F_{∞}. If F_{∞}, F_{1}, …, F_{q} are pairwise isomorphic, then F_{∞} is a Kantor/Knuth semifield flock.

This theorem as stated here is part of a more general result of Lunardon (1991). That theorem was originally given by Johnson, Lunardon and Wilke (1991), but relied upon the Classification of Finite Simple Groups. Lunardon’s proof removed this requirement. The theorem we need from the following chapter is Theorem 6.2, which in the present context says that for a semifield flock F_{∞}, the q flocks F_{1}, …, F_{q} above are pairwise isomorphic. Hence, since the Penttila-Williams/Bader-Lunardon-Pinneri flock F(C) = F_{∞} is not a Kantor/Knuth semifield flock, the q flocks F_{1}, …, F_{q} are not isomorphic to the original semifield flock F_{∞}, and so there is no collineation of GQ(C) which maps [A(∞)] to another line through (∞), by the Fundamental Theorem 4.9. Thus the subgroup calculated above is in fact the full subgroup of AutGQ(C) fixing the point O = ((0, 0), 0, (0, 0)). Hence, for the Penttila-Williams/Bader-Lunardon-Pinneri q-clan C,

|AutGQ(C)| = 5.22.(q − 1).q.q^{5}.

We note that Penttila and Williams (2000) gave the calculation of the stabiliser of the translation ovoid of Q(4, 3^{5}), showing it to be E_{3^{10}} ⋊ (C_{22} ⋊ C_{5}), where E_{n} is an elementary abelian group of order n and C_{m} is a cyclic group of order m. The process of moving from the translation ovoid to the flock implies that the subgroup of AutGQ(C) fixing O = ((0, 0), 0, (0, 0)) has order 5.22.q.(q − 1), confirming our calculation above.

CHAPTER 6 BLT-sets

We come now to the main topic of this thesis, namely, BLT-sets. The many connections developed between flocks of the quadratic cone, elation generalised quadrangles and translation planes motivated anew the concentrated study of flocks. In attempting to construct new examples of flocks from known ones, Bader, Lunardon and Thas (1990) developed the method of derivation of flocks when q is odd, which constructs q flocks from a given one, not all of which need be isomorphic with the original flock. This process involves a certain set of q + 1 points of the nondegenerate quadric in PG(4, q), called a BLT-set, from which the q + 1 flocks of the derivation may be reconstructed. Consequently, we have a more general model in which to study flocks of the quadratic cone for q odd, and furthermore, this model is still embedded within a projective space.

The assumption that q be odd is very important, since the process of derivation requires the use of the polarity arising from the nondegenerate quadric Q(4, q), and we have seen that even dimensional quadrics when q is even do not admit polarities. This assumption is somewhat clarified by considering Q(4, q) as a generalised quadrangle. Hence the rest of this thesis is devoted to the study of flocks and their associated structures over fields of odd characteristic only. The case for q even, although we have seen shares much of the same foundational material with q odd, is very different in nature when studied closer. Here there are associated herds of q + 1 ovals, and the techniques involved vary greatly from those for odd characteristic. As well as the linear flocks and the infinite families of examples connected with the Fisher-Thas/Walker flocks (Section 3.1) and the Payne monomial q-clans (Section 4.2), there exist two further infinite families called Subiaco (Cherowitzo, Penttila, Pinneri and Royle (1996)) and Adelaide (Cherowitzo, O’Keefe and Penttila (to appear)). Indeed, Cherowitzo, O’Keefe and Penttila (to appear) have given a unified construction of all of these families except that of Payne, and there currently exist no further examples in characteristic 2. As we shall see later, this is a very different situation to that of characteristic not 2. As well as the recent surveys mentioned at the beginning of Chapter 4, the case for q even is treated in detail in The Subiaco notebook of Payne [86].

6.1 Derivation of flocks and BLT-sets

Let F_{0} = {π_{1}, …, π_{q}} be a flock of the quadratic cone K_{0} with vertex P_{0} of PG(3, q), with q odd. We may embed K_{0} in a nondegenerate quadric Q of PG(4, q) so that K_{0} is the intersection with Q of the polar hyperplane H_{0} of P_{0} with respect to Q. For each plane π_{i} of F_{0}, denote by l_{i} the polar line of π_{i} with respect to Q. Then l_{i} is a secant line and meets Q in the points P_{0} and P_{i}, say. The polar hyperplane H_{i} of P_{i} with respect to Q meets Q in a quadratic cone K_{i} with vertex P_{i}. For any pair (i, j), i, j ∈ {1, …, q} with i ≠ j, denote by π_{ij} the plane H_{i} ∩ H_{j}, and let π_{jj} = π_{j}. By imposing coordinates upon PG(4, q) and considering the condition on planes of a flock (see Theorem 3.4), Bader, Lunardon and Thas (1990) proved the following.

Theorem 6.1 (Bader, Lunardon and Thas (1990)). For any j ∈ {1, …, q}, the set F_{j} = {π_{ij} ∩ K_{j}|i = 1, …, q} is a flock of the quadratic cone K_{j}.

The flocks F_{1}, …, F_{q} are said to be derived from the flock F_{0}.

Now consider a set {P_{0}, P_{1}, …, P_{q}} of q + 1 points of the nondegenerate quadric Q of PG(4, q), q odd, and let H_{i} be the polar hyperplane of P_{i} with respect to Q. Then the q planes H_{0} ∩ H_{i}, i ∈ {1, …, q}, will define by intersection a flock of the quadratic cone K_{0} = H_{0} ∩ Q if and only if the line H_{0} ∩ H_{i} ∩ H_{j} is exterior to Q for all i, j ∈ {1, …, q} with i ≠ j.

Conversely, for any point P_{0} of Q, let H_{0} denote the polar hyperplane of P_{0} with respect to Q, and let F_{0} be any flock of the quadratic cone K_{0} = H_{0} ∩ Q, so that F_{0} = {π_{1}, …, π_{q}} for some planes π_{1}, …, π_{q} of H_{0}. There exist q uniquely determined points P_{1}, …, P_{q} such that π_{i} = H_{0} ∩ H_{i} for all i ∈ {1, …, q}, where H_{i} is the polar hyperplane of P_{i} with respect to Q. Since F_{0} is a flock, the line H_{0} ∩ H_{i} ∩ H_{j} is exterior to Q for all i, j ∈ {1, …, q} with i ≠ j.

So starting with any flock F_{0} of the quadratic cone K_{0} = H_{0} ∩ Q, let P_{0}, P_{1}, …, P_{q} be the points associated with F_{0} as before. For any j ∈ {1, …, q}, the plane H_{0} ∩ H_{j} and the planes H_{i} ∩ H_{j} with i ∈ {1, …, q} such that i ≠ j, define a flock of the quadratic cone K_{j} = H_{j} ∩ Q by the above theorem. Hence any three distinct elements of {H_{0}, H_{1}, …, H_{q}} intersect in a line which is exterior to Q. As H_{i} is the polar hyperplane of P_{i}, any three distinct elements of {P_{0}, P_{1}, …, P_{q}} define a plane whose polar line is exterior to Q. Such a set of q + 1 points of Q = Q(4, q), q odd, is called a BLT-set, the name being due to Kantor (1991).

Example (The classical BLT-sets) Let l be an external line to Q and π = l^{⊥} be the polar plane to l with respect to Q. Then π ∩Q is a conic of π, so |π ∩Q| = q +1. Moreover, for distinct P, Q, R ∈ π ∩ Q, π = ⟨P, Q, R⟩, so ⟨P, Q, R⟩^{⊥} is an external line, namely l. Hence π ∩ Q is a BLT-set, called the classical BLT-set. Note that classical BLT-sets therefore constitute one isometry class of conics within Q(4, q), alluded to on page 14, the other class being those conics whose polar line is secant to Q(4, q). The classical BLT-set gives a linear flock at each point P ∈ π ∩ Q, because all the planes P^{⊥} ∩ Q^{⊥} lie on the line l = π^{⊥}.

If the flock F gives rise to a semifield or likeable translation plane via the André/Bruck-Bose constructions, then F is called a semifield or likeable flock, respectively.

Theorem 6.2 (Bader, Lunardon and Thas (1990)). If F_{0} is a semifield or likeable flock, then the flocks F_{1}, …, F_{q} derived from F_{0} are pairwise isomorphic.

Hence for semifield and likeable flocks, there can be at most one new class of flocks arising from the process of derivation. In general, Bader, Lunardon and Thas (1990) gave the following result regarding the set of flocks derived from a given BLT-set.

Theorem 6.3 (Bader, Lunardon and Thas (1990)). Let P_{1} and P_{2} be two points of the BLT-set P. Then the flock F_{1} arising from P_{1} is isomorphic to the flock F_{2} arising from P_{2} if and only if P_{1} and P_{2} are in the same orbit of P Γ O(5, q)_{P}.

Thus we have

Corollary 6.1. The number of distinct, non-isomorphic flocks arising from a BLTset P is equal to the number of orbits of P Γ O(5, q)_{P} on P.

Now note that the property defining a BLT-set is a condition on triples of points from the set. Thus we may define a partial BLT-set as a set of b ≥ 3 points of the nondegenerate quadric Q(4, q), q odd, such that any three distinct points in the set span a plane whose polar line with respect to Q(4, q) is an external line. Clearly, a partial BLT-set of size b will give rise to b partial flocks, that is, to b sets of pairwise disjoint conics on the relevant quadratic cones, each of size b − 1. We will investigate this notion a little further here, but the real power of this observation will become clear when we later use the computer to search for new examples of BLT-sets. Note that the condition for a set of points of Q(4, q) to be a BLT-set is equivalent to requiring that for any three points of the set, there is no point of Q(4, q) collinear (in a line of Q(4, q)) with all three of the points.

Since the nondegenerate quadric Q(4, q) of PG(4, q) is also a generalised quadrangle of order q, we consider the concept of a BLT-set in the GQ model of Q(4, q).

Motivated by the definition of a BLT-set in the quadric model above, we define a partial BLT-set of the generalised quadrangle Q(4, q) as a set P of b ≥ 3 points of Q(4, q) such that no point of Q(4, q) is collinear with more than 2 points of P. Note that we make no requirement upon the parity of q at this point, merely for purposes of investigating this definition in the abstract model of Q(4, q) as a generalised quadrangle.

Clearly, no two points of a partial BLT-set are collinear, since by the GQ axiom, a point of a BLT-set not on a line through two other points of the BLT-set will be collinear with some point on that line. Furthermore, consider a line l meeting the partial BLT-set P in a point P. By the GQ axiom, every point of P \{P } is collinear with a unique point of l. But no two points Q, R of P \ {P } can be collinear with the same point of l, since this common point would then be collinear with three points P, Q, R of P. Hence a partial BLT-set can have at most q + 1 points, the number of points on a line of Q(4, q). Thus we define a BLT-set of Q(4, q) to be a partial BLT-set of maximal size q + 1 (in accordance with our earlier definition in the quadric model).

Now consider a line l not meeting a BLT-set P. (Such a line l must exist : take a line m meeting P in a point P; then m contains a further point R not in P; as Q(4, q) is a GQ of order q ≥ 2, there are at least two further lines on R other than m; at most one of these further lines can meet P; and so we have a third line l on R not meeting P.) Consider a point R of l, and suppose R is collinear in the line m with a point P of P. Then we have seen earlier that each point of P is collinear with a distinct point of m, and so there is a second point Q of P collinear with the point R. Thus each point of a line not meeting a BLT-set is collinear with exactly 0 or 2 points of the BLT-set. Hence, with respect to the points of a line not meeting a BLT-set we may pair up the points of the BLT-set, and so the size of a BLT-set is divisible by 2, i.e., q + 1 is even. Thus BLT-sets can exist only when q is odd, as was required earlier in the quadric model of Q(4, q).

A set of points of a GQ such that every point not in the set is collinear with exactly 0 or 2 points of the set has been called a (0, 2)-set by De Soete and Thas (1984). Hence a BLT-set is a (0, 2)-set, and some of the above ideas and results, for example, the size of a BLT-set and the parity of q for the existence of a BLT-set, can also be found in their paper. Furthermore, the concept of BLT-set has been generalised to higher polar rank polar spaces, in a number of different ways. See, for example, O’Keefe and Thas (1997), Bader, O’Keefe and Penttila (submitted) and Shult (1996) (surveying joint work with Thas). In this thesis we will only be concerned with BLT-sets and partial BLT-sets of Q(4, q).

In Section 1.2 we defined the discriminant for nondegenerate orthogonal spaces with q odd, and saw that it provides a complete invariant of such spaces. Here we use the discriminant to study BLT-sets. The following results are due to Bader, O’Keefe and Penttila (submitted), the treatment of which we follow closely here, with the final corollary originally being due to Johnson (1992).

Let F = GF (q), q odd, let V = F^{5} be a vector space of rank 5 over F, and let Q be a nondegenerate quadratic form with polar form f defined on V. Then (V, Q) = O(5, q) is a nondegenerate orthogonal space, which defines a nondegenerate parabolic quadric Q(4, q) in the projective space PG(4, q) arising from V. Let ⊥ be the polarity arising from f. Lemma 6.1 (Bader, O’Keefe and Penttila (submitted)). If x, y, z ∈ O(5, q) are pairwise linearly independent singular vectors, then ⟨x, y, z⟩^{⊥} defines an external line with respect to the Q(4, q) if and only if

\frac{−2f(x, y)f(x, z)f(y, z)□}{disc(Q)} = ⊠ ∈ F^{∗}/□.

Proof : Since x, y, z are pairwise linearly independent singular vectors, W = ⟨x, y, z⟩ is a nondegenerate subspace of O(5, q). Let W^{⊥} = ⟨u, v⟩. Then S = {x, y, z, u, v} is a basis for V. Let B be the matrix of f with respect to S, and let B_{W} be the matrix of f restricted to W. Then

det(B) = det(B_{W}) det(B_{W^{⊥}}).

So

\frac{□}{disc(Q |_{W^{⊥}})} = \frac{disc(Q |_{W})}{disc(Q)}.

Now det(B_{W}) = 2f(x, y)f(x, z)f(y, z) since f is symmetric. So

\frac{□}{disc(Q |_{W^{⊥}})} = \frac{2f(x, y)f(x, z)f(y, z)□}{disc(Q)}.

Now consider the subspace W^{⊥}. This has rank 2, so is a line in PG(4, q). Hence W^{⊥} is an external line (an O^{−}(2, q) space) if and only if −disc(Q |_{W^{⊥}}) = ⊠. The result follows. Corollary 6.2. If P is a set of at least 3 points of Q(4, q), then P is a partial BLT-set if and only if for all ⟨x⟩, ⟨y⟩, ⟨z⟩ ∈ P we have:

\frac{−2f(x, y)f(x, z)f(y, z)□}{disc(Q)} = ⊠ ∈ F^{∗}/□.

Lemma 6.2 (Bader, O’Keefe and Penttila (submitted)). If P = {⟨x⟩, ⟨y⟩, ⟨z⟩, ⟨w⟩} is a set of four points of Q(4, q) such that {⟨x⟩, ⟨y⟩, ⟨z⟩}, {⟨x⟩, ⟨y⟩, ⟨w⟩} and {⟨x⟩, ⟨z⟩, ⟨w⟩} are partial BLT-sets, then P is a partial BLT-set.

Proof : It suffices to check that {⟨y⟩, ⟨z⟩, ⟨w⟩} is a partial BLT-set. Now we know that

−2f(x, y)f(x, z)f(y, z)□/disc(Q),
−2f(x, y)f(x, w)f(y, w)□/disc(Q),
−2f(x, z)f(x, w)f(z, w)□/disc(Q)

are all equal to ⊠ ∈ F^{∗}/□, hence their product

(\frac{−2f(x, y)f(x, z)f(x, w)□}{disc(Q)})^{2} \frac{−2f(y, z)f(y, w)f(z, w)□}{disc(Q)}

also equals ⊠ ∈ F^{∗}/□, and the result follows.

Corollary 6.3 (Johnson (1992)). Let P be a set of at least 3 points of Q(4, q). If there exists ⟨x⟩ ∈ P such that {⟨x⟩, ⟨y⟩, ⟨z⟩} is a partial BLT-set for all ⟨y⟩, ⟨z⟩ ∈ P \ {⟨x⟩}, then P is a partial BLT-set.

Hence to determine if a set of points of Q(4, q) is in fact a (partial) BLT-set, we need only consider the triples on one point of the set, and not all triples in the set. This observation, coupled with the previous corollary, provides a very cheap and easily implemented way for checking if a set of points of Q(4, q) is a (partial) BLT-set, when later we use the computer to search for new examples of BLT-sets.

6.2 BLT-sets, generalised quadrangles and translation planes

We have seen earlier that a flock gives rise to an EGQ of order (q^{2}, q) via a group coset geometry construction involving the associated q-clan. Alternatively, Knarr (1992) gave a beautiful geometric construction of this GQ directly from the associated BLT-set.

The construction of Knarr takes place in the polar space W(5, q), q odd, arising from a symplectic polarity of PG(5, q). This polar space has polar rank 3, and so contains points, lines and planes. The construction involves fixing a 3-dimensional subspace of PG(5, q) and considering the generalised quadrangle W(q) induced in this subspace. Observe that the duality between the generalised quadrangles W(q) and Q(4, q) (Theorem 1.12) means that the concept of a BLT-set of Q(4, q), q odd, can be transferred to W(q). Hence, a BLT-set of lines of W(q) is a set L of q + 1 lines of W(q) such that no line of W(q) is concurrent with more than 2 lines of L. A classical BLT-set of W(q) will then be a regulus of PG(3, q).

Let ρ be a symplectic polarity of PG(5, q), q odd, defining W(5, q), and fix a point P of W(5, q). Let Σ be a 3-dimensional subspace of PG(5, q) such that P ∉ Σ ⊂ P^{ρ}. In Σ, ρ induces a symplectic polarity, and hence a generalised quadrangle W(q). Let L = {l_{0}, l_{1}, …, l_{t}} be a BLT-set of W(q), and define the incidence structure K(L) as follows :

lines  points  lines  points
P  on P  collinear  not on P  not collinear
with P  with P
• ⨀ ⨀ ⨀ ⨀
lines m  totally isotropic
planes  of PG(5, q),  planes of ρ,  points
P π_{t} = P l_{t}  not on P,  not in P^{ρ},  of PG(5, q),
in one plane meeting some π_{t}  not in P^{ρ}
π_{t} = P l_{t}  in a line m

Incidence in K(L) is inherited from PG(5, q). Then we have that

Theorem 6.4 (Knarr (1992)). K(L) is a GQ of order (q^{2}, q).

Proof : We present as an example the proof of the GQ axiom for a nonincident point not collinear with P and line not on P. The order of the GQ follows from the combinatorics of W(5, q). We first note that a totally isotropic plane π (representing a line not on P) cannot meet two planes π_{s} = P l_{s}, π_{t} = P l_{t} (representing lines on P) in W(5, q), since the lines l_{s}, l_{t} are disjoint, being elements of a BLT-set. So suppose that Q is a point of PG(5, q) not in P^{ρ} (representing a point not collinear with P), and that π is a totally isotropic plane of ρ not in P^{ρ} meeting some plane π_{t} = P l_{t}, l_{t} ∈ L in a line n not on P (representing a line not on P), such that Q ∉ π. We must show that there is a unique totally isotropic plane π′ (of ρ not in P^{ρ} meeting some plane π_{s} = P l_{s}, l_{s} ∈ L in a line not on P) containing Q which meets π in a unique point Q′ of PG(5, q) (not in P^{ρ}). So, as Q is a point not on the maximal subspace π of W(5, q), by Tits’ polar space axiom, a = {x ∈ π|Q ∼ x} is a line of π. Project a from Q onto P^{ρ} to obtain the line b. Projecting b from P onto Σ gives a line c, which meets a line l ∈ L, and hence meets another line l′ ∈ L in the point X, as L is a (0, 2)-set with respect to lines. The point X corresponds to a point Y on the line b, and hence to a point Z on the line a in π. The line QZ is a line of W(5, q) which meets P^{ρ} in the point Y on the plane σ = P l′. The points on σ collinear with Q form the line m = {x ∈ σ|Q ∼ x}. Then the plane π′ = Qm and the point Z are the unique elements we require.

Furthermore, Knarr (1992) proved that K(L) is isomorphic to the generalised quadrangle arising from a flock associated to L via the group coset geometry construction. To see this, consider the following example.

Example (K(L) is an EGQ with base point P) Here let f((x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}), (y_{0}, y_{1}, y_{2}, y_{3}, y_{4}, y_{5})) = x_{0}y_{5} − x_{5}y_{0} + x_{1}y_{4} − x_{4}y_{1} + x_{2}y_{3} − x_{3}y_{2} be a nondegenerate alternating form on GF (q)^{6} defining the symplectic polarity ρ and so W(5, q). Let

G = \{M_{a,b,c,d,e} = (\begin{matrix}1 & a & b & c & d & e \\ 0 & 1 & 0 & 0 & 0 & d \\ 0 & 0 & 1 & 0 & 0 & c \\ 0 & 0 & 0 & 1 & 0 & −b \\ 0 & 0 & 0 & 0 & 1 & −a \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) |a, b, c, d, e ∈ GF (q)\}

Then G is a group of isometries of the vector space GF (q)^{6} with form f, so G ≤ Sp(6, q). Thus G commutes with ρ and gives a group of automorphisms of the polar space Π(ρ) =W(5, q). G fixes the point P (1, 0, 0, 0, 0, 0) and fixes every subspace of P^{ρ} on P. Hence, if we perform the Knarr construction using P, G will fix each plane that represents a line on P. Hence G fixes the set of lines that represent points collinear with P and the set of totally isotropic planes that represent lines not on P. G acts regularly on the set of points of PG(5, q) not in P^{ρ}, and so K(L) is an EGQ with base point P.

By multiplying two elements of G we see that

\begin{aligned}M_{a,b,c,d,e}M_{a′,b′,c′,d′,e′} & = & M_{a+a′,b+b′,c+c′,d+d′,e+e′+ad′+bc′−b′c−a′d} \\ & ≠ & M_{a′,b′,c′,d′,e′}M_{a,b,c,d,e}, \end{aligned}

and so G is a non-abelian group of order q^{5}, with centre

Z = Z(G) = {M_{0,0,0,0,e}|e ∈ GF (q)}

of order q, and G/Z is an elementary abelian group of order q^{4}. Furthermore, the centre Z is both the derived subgroup G′ and the Frattini subgroup Φ(G) : Z(G) = G′ = Φ(G). Indeed, this group G is isomorphic to the group of Payne in Section 4.2.

Note that K(L) is isomorphic to the classical GQ H(3, q^{2}) if and only if L is a classical BLT-set. As the Knarr construction relies on the existence of a BLT-set, we cannot use this construction for the case of q even. However, Thas (1999) has recently given a stunning geometric construction of these flock GQs which holds for all q. See Thas (1999) and also Thas (2001) [119] for this construction and the connections with the Knarr construction.

Working in the group coset geometry model of a GQ arising from a q-clan, Payne and Rogers (1990) were able to show that for any flock F, the GQ arising from any derived flock is a recoordinatised version of the GQ arising from F, and hence is isomorphic with the GQ arising from F. In fact, each flock derived from F corresponds to a line through the base point (∞) of the GQ arising from F, with F corresponding to the line [A(∞)], as was alluded to in the previous chapter. Hence derivation of F corresponds to interchanging the line [A(∞)] with each other line through (∞). In terms of the associated BLT-set, we then have the following.

Theorem 6.5 (Payne and Rogers (1990)). A BLT-set gives rise to exactly one generalised quadrangle.

Thus we see that BLT-sets are as general as GQs with respect to modelling flocks of the quadratic cone. As BLT-sets are substructures of polar spaces, two BLT-sets P_{1} and P_{2} of Q(4, q) (respectively, W (q)) are said to be isomorphic if and only if there exists an element of P Γ O(5, q) (respectively, P Γ Sp(4, q)) which maps P_{1} onto P_{2}. The following result was first given by Kantor (1991) in the context of the group coset geometry construction, but was also proved by Payne and Thas (1991)

[100] in a more geometric fashion.

Theorem 6.6. The BLT-sets P_{1} and P_{2} are isomorphic if and only if the generalised quadrangle arising from P_{1} is isomorphic to the generalised quadrangle arising from P_{2}.

Hence the automorphism group of the generalised quadrangle arising from a BLT-set can be calculated from the group of the BLT-set, by which we mean the stabiliser of the BLT-set in the automorphism group of the relevant polar space in which the BLT-set is defined, namely in P Γ O(5, q) or P Γ Sp(4, q). In particular, every collineation of the GQ arising from a BLT-set induces an automorphism of the space of the BLT-set which fixes the BLT-set. Since the points of the BLT-set correspond to the lines through (∞) of the associated GQ, the group of the BLT-set corresponds to the subgroup of the automorphism group of the associated GQ acting on the lines through (∞), modulo the kernel of this action (except in the case of the Kantor/Knuth semifield example, where some elements of the kernel do indeed induce nontrivial automorphisms of the space of the BLT-set (fixing the BLT-set)).

We have seen that a flock gives rise to many linespreads of PG(3, q), both via the Thas/Walker construction of an ovoid of the Klein quadric and via the construction of regular hyperbolic fibrations with constant back half. Thus we may interpret these constructions instead directly from the associated BLT-set.

The ovoid of the Klein quadric constructed from a flock by the Thas/Walker construction (Section 4.1.1) consists of a union of q conics pairwise intersecting in a common point. The conics are the intersection with the Klein quadric of the planes which are polar to each of the planes of the given flock, and the common point is the vertex of the underlying quadratic cone. Now embed the quadratic cone K of PG(3, q) as a hyperplane section of the nondegenerate parabolic quadric Q(4, q), which in turn is embedded as a section of the Klein quadric by the hyperplane H of PG(5, q). Let F be a flock of K and let P be the BLT-set of Q(4, q) associated to F. Then one point of P is the vertex V of K, whilst the other q points of P correspond to the planes of F. If Q is the polar point to H, then the ovoid arising from the Thas/Walker construction is the set of points of the quadric on the planes ⟨V, Q, P ⟩, where P is a point of P other than V. Hence for each choice of a point of a BLT-set of Q(4, q), we can construct an ovoid of the Klein quadric and hence a linespread of PG(3, q). Clearly points in the same orbit of the BLT-set will give rise to equivalent ovoids.

By the process of derivation, each point of a BLT-set determines a flock, each plane of which determines a regular hyperbolic fibration with constant back half (Section 4.4). Since each plane of a flock arising from a point P of a BLT-set P corresponds to a distinct point of P other than P, we see that each hyperbolic fibration arising from a flock corresponds to an ordered pair of points of the associated BLT-set. Hence equivalence of the regular hyperbolic fibrations with constant back half arising from a BLT-set P corresponds to orbits on ordered pairs of points of P under the stabiliser of P in P Γ O(5, q). (See Baker, Ebert and Penttila (submitted).)

6.3 The known infinite families of BLT-sets

Consider the process of derivation in an analytic setting, that is, with coordinates.

Let the nondegenerate quadric Q(4, q) of PG(4, q), q odd, be defined by the quadratic form Q : x_{0}x_{4} + x_{1}x_{3} − x_{2}^{2} = 0, and let H_{0} be the polar hyperplane of the point P_{0} = (0, 0, 0, 0, 1) with respect to Q(4, q). Then H_{0} meets Q(4, q) in the quadratic cone K_{0} with equations x_{0} = x_{1}x_{3} − x_{2}^{2} = 0, and P_{0} is the vertex of K_{0}. Let F_{0} be any flock of K_{0}, where the planes π_{i} = [a_{i}, b_{i}, c_{i}, 1]^{T} of F_{0} have equations x_{0} = a_{i}x_{1}+b_{i}x_{2}+c_{i}x_{3}+x_{4} = 0, i ∈ {1, …, q}, with (b_{i}−b_{j})^{2}−4(a_{i}−a_{j})(c_{i}−c_{j}) = ⊠, i ≠ j. Then we saw above that the polar line l_{i} of π_{i} with respect to Q(4, q) meets Q(4, q) in two points P_{0} and P_{i}. A short calculation shows that P_{i} = (1, c_{i}, −\frac{b_{i}}{2} , a_{i}, \frac{b_{i}^{2}}{4} −a_{i}c_{i}), and so given coordinates for planes of a flock we can write down the coordinates

of the associated BLT-set. In other words, if C = \{(\begin{matrix}a_{t} & \frac{1}{2}b_{t} \\ \frac{1}{2}b_{t} & c_{t} \end{matrix}) : t ∈ GF (q)\} is

a q-clan, then the associated BLT-set, with distinguished point (0, 0, 0, 0, 1), can be written in the following form, where Q(4, q) is defined by the quadratic form Q : x_{0}x_{4} + x_{1}x_{3} − x_{2}^{2}.

P(C) = \{(1, c_{t}, −\frac{b_{t}}{2} , a_{t}, \frac{b_{t}^{2}}{4} − a_{t}c_{t})|t ∈ GF (q)\} ∪ \{(0, 0, 0, 0, 1)\}.

An important consequence of the development of derivation of flocks is the possibility of obtaining new examples of flocks from known ones. Bader, Lunardon and Thas (1990) investigated this for each of the examples known at that time. We have seen many of the known examples in earlier sections as examples of one or another of the various related structures. Here we collect together these examples, and present them as q-clans corresponding to flocks of the quadratic cone given above, so that the corresponding BLT-set with distinguished point (0, 0, 0, 0, 1) is defined with respect to the quadratic form Q : x_{0}x_{4}+x_{1}x_{3}−x_{2}^{2}. The calculation of the q-clan condition (see Theorem 4.5) for each example is given here for consistency. For each family of BLT-sets we will discuss the various related structures and attempt to give complete references for where these appeared in the literature before the connections were known. Note that the paper of Gevaert and Johnson (1988) provided at that time a unified description of both the connections and the known examples. The associated ovoids, linespreads and translation planes discussed here will be those arising via the Thas/Walker and André/Bruck-Bose constructions.

The general group orders presented are taken from the survey of Payne (1992) of the collineation groups of the GQs known at that time. The calculations appeared in various papers predating Payne (1992), and we refer to this paper for full references. Here we have factored out the q^{5} elations and the q − 1 collineations in the kernel of the action on the lines through (∞) to obtain the order in general of the stabiliser P Γ O(5, q)_{P} of the BLT-set P in P Γ O(5, q), where q = p^{h}. Note that for the Kantor/Knuth semifield example, we do not factor out the entire kernel, since some of these collineations do induce automorphisms of the space of the BLT-set fixing the BLT-set, in particular, hyperbolic reflections with axis the hyperbolic quadric spanned by the BLT-set (see Corollary 6.5). For small q, some examples belong to more than one infinite family, which may or may not yet be known, and consequently the group order may grow somewhat in various cases. Using a computer, as we discuss in later chapters, we have calculated the group orders for all the known examples over small fields, and to clarify the general statements in Payne (1992), we here state explicitly the group order for each example of the infinite family, since this is a significant piece of information when studying small examples.

There exist two further families which we have not yet seen in detail, namely the family of Fisher (Fisher and Thas 1979) and the family of Penttila (1998). Since their description as q-clans is somewhat involved, these two examples will be discussed later in the alternative model of Q(4, q) given by Penttila (1998), in which the second example was originally described.

For clarity, we note here that there exists a further infinite family of BLT-sets, published by Penttila and the present author (2001) and appearing in this thesis in Chapter 9.

The classical BLT-sets

These have been described in the example on page 70, and exist for all odd q. If n is a fixed nonsquare of GF (q), then the classical q-clan is

C = \{(\begin{matrix}t & 0 \\ 0 & −nt \end{matrix}) : t ∈ GF (q)\}.

We saw that all flocks arising from the classical BLT-set are linear, and so all linespreads are regular. Hence the translation planes arising are Desarguesian. The generalised quadrangle arising is isomorphic to the classical GQ H(3, q^{2}) (see Payne and Thas (1984)), which has automorphism group P Γ U(4, q^{2}). In this case, the kernel of the GQ is larger than that of nonclassical GQs, and has size 2(q − 1)(q + 1).

|P Γ O(5, q)_{P}| = 2h(q − 1)q(q + 1)^{2}.

The Fisher-Thas/Walker/Kantor BLT-sets

For q odd and q ≡ −1 (mod 3) :

C = \{(\begin{matrix}t & \frac{3}{2}t^{2} \\ \frac{3}{2}t^{2} & 3t^{3} \end{matrix}) : t ∈ GF (q)\}

(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t})

= (3s^{2} − 3t^{2})^{2} − 4(s − t)(3s^{3} − 3t^{3})
= 9(s − t)^{2}(s + t)^{2} − 12(s − t)^{2}(s^{2} + st + t^{2})
= (s − t)^{2}(−3s^{2} + 6st − 3t^{2})
= −3(s − t)^{4}

Since q ≡ −1 (mod 3), −3 is a non-square in GF (q).

The Fisher-Thas/Walker flock described on page 43 is the flock arising from the point (0, 0, 0, 0, 1) of this BLT-set. In fact, this flock gives rise to a likeable translation plane, where the likeable function is just the zero function L(t) = 0 for all t ∈ GF (q). Hence by Theorem 6.2, all the derived flocks are isomorphic. Bader, Lunardon and Thas (1990) showed that the flock arising from the point (1, 0, 0, 0, 0) is isomorphic to the original flock, and hence all flocks arising from this BLT-set are Fisher-Thas/Walker flocks. Hence the translation planes arising are those of Walker (1976). Note that these examples can be defined similarly for q even and q ≡ −1 (mod 3), q > 2, as we have seen. In this case, the associated translation planes were first described by Betten (1973). The generalised quadrangle arising in general was first given by Kantor (1980), from a construction involving the G_{2}(q) generalised hexagon. We note that it was this paper which began the detailed study of generalised quadrangles as group coset geometries.

|P Γ O(5, q)_{P}| = \begin{cases}h(q − 1)q(q + 1), q > 5; \\ 720, q = 5. \end{cases}

Theorem 6.7 (Thas (1993)). Let F be a flock of the quadratic cone of PG(3, q). Assume q ≥ 4 and, if q is odd, assume that

q ∉ {17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83}.

Then F is the flock of Fisher-Thas/Walker if and only if no four planes of F have a point in common.

A (q + 1)-arc of PG(4, q) is a set of q + 1 points such that any 5 points of the set span the whole space.

Corollary 6.4. With the same conditions on q as in the previous theorem, a set of q + 1 points of Q(4, q) is a Fisher-Thas/Walker/Kantor BLT-set if and only if it is a (q + 1)-arc.

The Kantor/Knuth semifield BLT-sets

For q odd and not prime, n a fixed nonsquare of GF (q) and σ ≠ 1 a nonidentity automorphism of GF (q) :

C = \{(\begin{matrix}t & 0 \\ 0 & −nt^{σ} \end{matrix}) : t ∈ GF (q)\}

(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t})

= (0 − 0)^{2} − 4(s − t)(−ns^{σ} + nt^{σ})
= 4n(s − t)(s^{σ} − t^{σ})
= 4n(s − t)(s − t)^{σ}
= 4n(s − t)^{σ+1}

Since q is odd, if x ↦ x^{σ} is an automorphism of GF (q) then σ is odd, and so σ + 1 is even.

The Kantor/Knuth semifield ovoid described on page 50 with τ = 1 is the ovoid arising from the point (0, 0, 0, 0, 1) of this BLT-set. Hence the associated flock is a semifield flock, and so by Theorem 6.2, all the derived flocks are isomorphic. Bader, Lunardon and Thas (1990) showed that the flock arising from the point (1, 0, 0, 0, 0) is isomorphic to the original flock, and so all flocks arising from this BLT-set are isomorphic semifield flocks. The generalised quadrangle arising was first given by Kantor (1986) in the group coset geometry model, before the connections with ovoids and translation planes were known. Note that equivalent BLT-sets arise from different field automorphisms σ, γ exactly when γ = σ^{±1}.

|P Γ O(5, q)_{P}| = \begin{cases}2h(q − 1)q(q + 1), σ^{2} ≠ 1; \\ 4h(q − 1)q(q + 1), σ^{2} = 1 ≠ σ. \end{cases}

Theorem 6.8 (Thas (1987)). If all planes of a nonlinear flock F intersect in a common point, then F arises from a Kantor/Knuth semifield BLT-set.

Let P be the Kantor/Knuth semifield BLT-set arising from the q-clan above. Then the point common to all planes of the flock F arising from the point (0, 0, 0, 0, 1) of P is the point P = (0, 0, 1, 0, 0), which is not on Q(4, q). Since P lies on every plane of F, every point of P will lie in the polar hyperplane H of P. Hence we have the following corollary.

Corollary 6.5. The points of a Kantor/Knuth semifield BLT-set span a hyperplane of PG(4, q) meeting Q(4, q) in a hyperbolic quadric Q^{+}(3, q).

The Kantor monomial BLT-sets

For q odd and q ≡ ±2 (mod 5) :

C = \{(\begin{matrix}t & \frac{5}{2}t^{3} \\ \frac{5}{2}t^{3} & 5t^{5} \end{matrix}) : t ∈ GF (q)\}

(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t})
= (5s^{3} − 5t^{3})^{2} − 4(s − t)(5s^{5} − 5t^{5})
= 25(s − t)^{2}(s^{2} + st + t^{2})^{2} − 20(s − t)^{2}(s^{4} + s^{3}t + s^{2}t^{2} + st^{3} + t^{4})
= (s − t)^{2}(5s^{4} + 30s^{3}t + 55s^{2}t^{2} + 30st^{3} + 5t^{4})
= 5(s − t)^{2}(s^{2} + 3st + t^{2})^{2}

Since q ≡ ±2 (mod 5), 5 is a nonsquare in GF (q).

The Kantor monomial q-clan on page 52 with q odd is the q-clan given here. Bader, Lunardon and Thas (1990) showed that the flocks arising from the points (0, 0, 0, 0, 1) and (1, 0, 0, 0, 0) of this BLT-set are isomorphic, and that the remaining q − 1 flocks are also pairwise isomorphic. This was done by considering the associated translation planes, which Bader (1988) [6] shows are equivalent to the planes of Narayana Rao, Satyanarayana and Vithal Rao (1987) in the case q ≢ 0 (mod 3). The translation planes are also related to some planes of Cohen and Ganley (1984) and of Barriga (1987). Both Johnson (1992) and Payne (1992) then showed that indeed the remaining q − 1 derived flocks were not isomorphic with the first two flocks, and so exactly two classes of nonisomorphic flocks arise from this BLT-set.

|P Γ O(5, q)_{P}| = \begin{cases}2h(q − 1), q > 13; \\ 48,  q = 13; \\ 384,  q = 7; \\ 192,  q = 3. \end{cases}

The Kantor likeable BLT-sets

For q = 5^{h} and n a fixed non-square of GF (q) :

C = \{(\begin{matrix}t & 3t^{2} \\ 3t^{2} & n^{−1}t + 2t^{3} + nt^{5} \end{matrix}) : t ∈ GF (q)\}

(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t})
= (s^{2} − t^{2})^{2} − 4(s − t)(n^{−1}s + 2s^{3} + ns^{5} − n^{−1}t − 2t^{3} − nt^{5})
= (s − t)^{2}(s + t)^{2} − 4n^{−1}(s − t)^{2}(1 + 2n(s^{2} + st + t^{2})
+n^{2}(s^{4} + s^{3}t + s^{2}t^{2} + st^{3} + t^{4}))
= n^{−1}(s − t)^{2}(−4 + n(−7s^{2} − 6st − 7t^{2}) − 4n^{2}(s^{4} + s^{3}t + s^{2}t^{2} + st^{3} + t^{4}))
= n^{−1}(s − t)^{2}(1 − 2n(s^{2} − 2st + t^{2}) + n^{2}(s^{4} + s^{3}t + s^{2}t^{2} + st^{3} + t^{4}))
= n^{−1}(s − t)^{2}(1 − 2n(s − t)^{2} + n^{2}(s − t)^{4})
= n^{−1}(s − t)^{2}(1 − n(s − t)^{2})^{2}

The Kantor likeable translation plane on page 47 is the translation plane corresponding to the flock arising from the point (0, 0, 0, 0, 1) of this BLT-set. Hence this flock is likeable, and so by Theorem 6.2, all the derived flocks are isomorphic. Bader, Lunardon and Thas (1990) showed that in fact the derived flocks were not isomorphic to the original flock, and so exactly two classes of nonisomorphic flocks arise from this BLT-set.

|P Γ O(5, q)_{P}| = \begin{cases}2hq, q > 5; \\ 720, q = 5. \end{cases}

The Ganley semifield BLT-sets

For q = 3^{h} and n a fixed nonsquare of GF (q) :

C = \{(\begin{matrix}t & −t^{3} \\ −t^{3} & −nt − n^{−1}t^{9} \end{matrix}) : t ∈ GF (q)\}

(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t})
= (s^{3} − t^{3})^{2} − 4(s − t)(−ns − n^{−1}s^{9} + nt + n^{−1}t^{9})
= (s − t)^{2}(s^{2} + st + t^{2})^{2}
+4n^{−1}(s − t)^{2}(n^{2} + s^{8} + s^{7}t + s^{6}t^{2} + s^{5}t^{3} + s^{4}t^{4} + s^{3}t^{5} + s^{2}t^{6} + st^{7} + t^{8})
= n^{−1}(s − t)^{2}(4n^{2} + n(s^{2} + st + t^{2})^{2}
+4(s^{8} + s^{7}t + s^{6}t^{2} + s^{5}t^{3} + s^{4}t^{4} + s^{3}t^{5} + s^{2}t^{6} + st^{7} + t^{8}))
= n^{−1}(s − t)^{2}(n^{2} − 2n(s^{2} + st + t^{2})^{2}
+(s^{8} + s^{7}t + s^{6}t^{2} + s^{5}t^{3} + s^{4}t^{4} + s^{3}t^{5} + s^{2}t^{6} + st^{7} + t^{8}))
= n^{−1}(s − t)^{2}(n − (s^{2} + st + t^{2})^{2})^{2}

The plane coordinatised by the Ganley semifield on page 31 is the translation plane corresponding to the flock arising from the point (0, 0, 0, 0, 1) of this BLT-set. Hence this flock is semifield, and so by Theorem 6.2, all the derived flocks are isomorphic. Both Payne and Thas (1991) [99] and Johnson, Lunardon and Wilke (1991) showed that in fact the derived flocks were not isomorphic to the original flock, and so exactly two classes of nonisomorphic flocks arise from this BLT-set. In particular, the derived flocks are not semifield.

|P Γ O(5, q)_{P}| = \begin{cases}2hq, q > 27; \\ 648, q = 27; \\ 5760, q = 9; \\ 192, q = 3. \end{cases}

6.3.1 An alternative model of Q(4, q) As mentioned above, there exist two further infinite families of BLT-sets. The family of flocks due to Fisher (Fisher and Thas (1979)) can be described in a number of different ways, though these are all somewhat involved. In particular, the explicit description in terms of q-clans, due to Payne (1988), can be given as follows.

For all odd q, let ζ be a primitive element of GF (q^{2}), so ω = ζ^{q+1} is a primitive element of GF (q) and hence a nonsquare of GF (q). Put i = ζ^{(q+1)/2}, so i^{2} = ω, i^{q} = −i. Put z = ζ^{q−1} = a + bi, so z has order q + 1 in the multiplicative group of GF (q^{2}). Then the matrices A_{t} of the q-clan C are given by:

A_{t} = [ \begin{matrix}t & 0 \\ 0 & −ωt \end{matrix} ]

for those t ∈ GF (q) with t^{2} − 2(1 + a)^{−1} a square in GF (q), and the other matrices are those of the form

A_{t} = [ \begin{matrix}−a_{2j} & b_{2j} \\ b_{2j} & −ωa_{2j} \end{matrix} ]

for 0 ≤ j ≤ (q − 1)/2 and with

a_{k} = (z^{k+1} + z^{−k})/(z + 1)
b_{k} = i(z^{k+1} − z^{−k})/(z + 1).

Recall from Section 3.1 that the Fisher flocks have been characterised as follows.

Theorem 6.9 (Payne and Thas (1991) [99]). If F is a nonlinear flock of the quadratic cone for which \frac{q−1}{2} of the planes of F intersect in a common line, then F is a Fisher flock.

Let F be a flock of Fisher type of the quadratic cone K embedded in Q(4, q), and let P be the associated BLT-set. Then \frac{q−1}{2} of the planes of F meet in a common line l, which is exterior to Q(4, q). Hence the \frac{q−1}{2} points of P corresponding to these planes lie on the plane π = l^{⊥}, which also contains the vertex of K. Now π intersects Q(4, q) in a conic, which is a classical BLT-set since the polar line of π is an external line, and so \frac{q+1}{2} of the points of a Fisher BLT-set meet some classical BLT-set. By the characterisation of Payne and Thas (1991) above, the Fisher BLT-set is the only BLT-set meeting a classical BLT-set in \frac{q+1}{2} points.

Bader (1990) and Payne and Thas (1991) [99] showed that all flocks derived from a Fisher flock are equivalent to a Fisher flock, and so the associated BLT-set is transitive, that is, the stabiliser of the Fisher BLT-set in P Γ O(5, q) acts transitively on the set of points of the BLT-set. From Payne (1992) and our computer calculations discussed above, the order of the stabiliser of a Fisher BLT-set is

|P Γ O(5, q)_{P}| = \begin{cases}2h(q + 1)^{2}, q > 7; \\ 384,  q = 7; \\ 720,  q = 5; \\ 192,  q = 3. \end{cases}

Payne (1988) also showed that the translation planes of Baker and Ebert (1988) correspond to some of the translation planes arising from the Fisher flocks.

We now present an alternative model of Q(4, q) which was introduced by Penttila (1998). Within this model Penttila describes the classical BLT-sets, the Fisher-Thas/Walker/Kantor BLT-sets and the Fisher BLT-sets, and also gives a new family of BLT-sets. We will give a proof that the Fisher BLT-sets presented are indeed those arising from the flocks of Fisher (Fisher and Thas (1979)), and present the new family of examples due to Penttila (1998).

Let V = {(x, y, a) : x, y ∈ GF (q^{2}), a ∈ GF (q)}, considered as a vector space of rank 5 over GF (q), q odd. Define the following maps

- : GF (q^{2}) → GF (q^{2}) by x̄ = x^{q} ∀x ∈ GF (q^{2})
N : GF (q^{2}) → GF (q) by N(x) = xx̄ ∀x ∈ GF (q^{2})
T : GF (q^{2}) → GF (q) by T(x) = x + x̄ ∀x ∈ GF (q^{2})
Q : V → GF (q) by Q(x, y, a) =N(x)+N(y) + a^{2} ∀(x, y, a) ∈ V

f : V × V → GF (q) by f((x_{1}, y_{1}, a_{1}), (x_{2}, y_{2}, a_{2})) =T(x_{1}x̄_{2})+T(y_{1}ȳ_{2}) + 2a_{1}a_{2} ∀(x_{1}, y_{1}, a_{1}), (x_{2}, y_{2}, a_{2}) ∈ V

so that − is conjugation, N is the relative norm, T is the relative trace, and Q is a non-degenerate quadratic form on V which polarises to the bilinear form f. Thus

Q_{1} = {⟨(x, y, a)⟩|Q(x, y, a) = 0}

is a parabolic quadric of the projective space PG(4, q) arising from V.

The classical BLT-sets are of the form {⟨(x, 0, 1)⟩ : x ∈ GF (q^{2}), N(x) = −1}.

Before presenting the Fisher BLT-sets, we make some useful observations.

(1) T(α^{2}) = T(α)^{2} − 2N(α) ∀α ∈ GF (q^{2}), since T(α)^{2} = (α + ᾱ)^{2} = α^{2} + ᾱ^{2} + 2αᾱ =T(α^{2}) + 2N(α)

(2) T(α^{2}) − 2 = ⊠ ∀α ∈ GF (q^{2}) with N(α) = 1 and α ≠ ±1, since, by (1), T(α^{2}) − 2 =T(α)^{2} − 4, which is the discriminant of the minimal polynomial x^{2}−T(α)x + 1 of α, which is a non-square, as the only elements of GF (q) of norm 1 are ±1.

(3) Since V is the orthogonal direct sum of {(x, 0, 0)|x ∈ GF (q^{2})}, {(0, y, 0)|y ∈ GF (q^{2})} and {(0, 0, a)|a ∈ GF (q)}, the discriminant of Q is the product of the discriminants of Q restricted to these subspaces. The first two subspaces are isometric, and so have the same discriminant, whereas the discriminant of Q restricted to the third subspace is clearly 2. Hence disc(Q) = 2□.

Example (The Fisher BLT-sets) For any odd q, fix β ∈ GF (q^{2}) with β ≠ 1, N(β) = −1. Let

P = {⟨(βx^{2}, 0, 1)⟩|N(x) = 1} ∪ {⟨(0, βy^{2}, 1)⟩|N(y) = 1}.

Then P is a BLT-set of Q_{1}, due to Fisher (Fisher-Thas (1979)). We prove this as follows.

Define P_{1} = {⟨(βx^{2}, 0, 1)⟩|N(x) = 1} and P_{2} = {⟨(0, βy^{2}, 1)⟩|N(y) = 1}. Then Q(βx^{2}, 0, 1) = N(β)N(x^{2}) + 1 = 0 and Q(0, βy^{2}, 1) = N(β)N(y^{2}) + 1 = 0, so P = P_{1} ∪ P_{2} ⊆ Q_{1} of size q + 1.

Let ⟨v⟩ = ⟨(βx_{1}^{2}, 0, 1)⟩, N(x_{1}) = 1 and ⟨w⟩ = ⟨(βx_{2}^{2}, 0, 1)⟩, N(x_{2}) = 1 be distinct points of P_{1}. Then f(v, w) = f((βx_{1}^{2}, 0, 1), (βx_{2}^{2}, 0, 1)) = T(−x_{1}^{2}x̄_{2}^{2}) + 2 = −T(x_{1}^{2}x̄_{2}^{2}) + 2. Let α = x_{1}x̄_{2}. Then N(α) = 1. Since ⟨v⟩ ≠ ⟨w⟩, α^{2} ≠ 1. By (2), T(α^{2}) − 2 = ⊠, and so we have f(v, w) = −⊠ for any two distinct points of P_{1}. Similarly for P_{2}. Clearly, if ⟨v⟩ is a point of P_{1} and ⟨w⟩ is a point of P_{2}, then f(v, w) = 2. Hence for any three distinct points ⟨u⟩, ⟨v⟩, ⟨w⟩ in P = P_{1} ∪ P_{2}, we have

f(u, v)f(u, w)f(v, w) = − ⊠ .

By (3), the discriminant of Q equals 2□, and so by Corollary 6.2, P is a BLT-set of Q_{1}. Since P meets the classical BLT-set given above in exactly \frac{q+1}{2} points, by Theorem 6.9 we have that P is equivalent to a BLT-set of Fisher.

The maps (x, y, a) ↦ (y, x, a), (x, y, a) ↦ (λ^{2}x, y, a) (N(λ) = 1) and (x, y, a) ↦ (x, λ^{2}y, a) (N(λ) = 1) are isometries of (V, Q) stabilising P. Furthermore, they act transitively on P, and so we see clearly that all flocks derived from a Fisher flock are again Fisher flocks, as shown by Bader (1990) and Payne and Thas (1991) [99].

We now present the family of BLT-sets given by Penttila (1998). A proof that they are indeed new BLT-sets appears in that paper. They had not been previously described in any of the associated models.

Example (The Penttila BLT-sets) For q ≡ ±1 (mod 10), fix β, γ ∈ GF (q^{2}) with N(β) = −\frac{4}{5} and N(γ) = −\frac{1}{5}. Let

P = {⟨(βx^{2}, γx^{3}, 1)⟩|N(x) = 1}.

Then P is a BLT-set of Q_{1} due to Penttila (1998).

If λ is an element of GF (q^{2}) of order q + 1 with N(λ) = 1, the cyclic group of order q + 1 defined by the isometry (x, y, a) ↦ (λ^{2}x, λ^{3}y, a) of (V, Q) stabilises P. In fact, this group acts regularly on P, and so there is only one equivalence class of flocks arising. The full group of this example has not yet been calculated.

90  Chapter 6. BLT-sets

CHAPTER 7 Searching for BLT-sets

We have now seen that BLT-sets provide a general model for studying flocks of the quadratic cone and their associated geometries over fields of odd characteristic. Indeed, one BLT-set may give rise to a number of nonisomorphic flocks, each of which gives rise to a linespread of PG(3, q) via the Thas/Walker construction, as well as a number of linespreads via the associated regular hyperbolic fibrations. Each of these linespreads then admits a number of infinite families of linespreads via the construction of Hiramine, Matsumoto and Oyama. Hence a single BLT-set can give rise to many infinite families of translation planes of order q^{2} and dimension two over their kernel. As we have seen, only the generalised quadrangle model can provide as general a setting as that of BLT-sets. An added feature of the BLT-set model is that it is embedded in projective space, unlike the GQ model. These observations are particularly relevant with respect to the central goal of this project, which is to search for new examples of these interconnected structures using a computer.

As well as the infinite families of BLT-sets described in the previous chapter, we saw earlier the q-clan of Bader, Lunardon and Pinneri for q = 243, arising from the translation ovoid of Penttila and Williams. There exist further individual examples of BLT-sets, not yet belonging to infinite families, which have also been found with the use of computers. De Clerck and Herssens (1992) used the computer algebra packages CAYLEY and GAP to construct flocks of the quadratic cone, finding one example for q = 11, one example for q = 16, two examples for q = 17, related by derivation, and two examples for q = 23, also related by derivation. Penttila and Royle (1998) then extended these searches by considering the more general setting of BLT-sets, and using back-track algorithms with the collinearity graph of Q(4, q) as a GQ, found one example for q = 17, three examples for q = 19, four examples for q = 23 and two examples for q = 25. These new BLT-sets gave rise to 26 new flocks. An unpublished result of De Clerck and Penttila (1997) described a further example of a flock for q = 47.

One of the most important reasons for attempting to describe individual examples of BLT-sets (or flocks or GQs or translation planes) is the possibility of constructing infinite families by way of analysis of these examples. Indeed, the example of De Clerck and Herssens (1992) for q = 11, denoted by them as DCHT (11), was generalised to an infinite family of partial flocks of size 11 for any q ≡ −1 mod 12 in a computerfree construction by Thas, Herssens and De Clerck (1993). Furthermore, the BLT-set arising from the flock DCHT (11) and the example of Penttila and Royle (1998) for q = 19 with a group of order 40 are examples of the infinite family of BLT-sets of Penttila (1998) in Section 6.3.1. The wealth of existing data suggests the possibility of more infinite families, and yet any new family discovered may only contain one or two of these existing examples. Thus we feel that continuing these sorts of experimental searches, in particular, continuing past the field of order 25 where Penttila and Royle left off, is of worthwhile value. Hence the main aim of this thesis is to search for new examples of BLT-sets using a computer. Advances in both theoretical and computational aspects of this problem have made this a viable possibility, and in this chapter we discuss the ideas behind these searches.

Note that, although thorough, the searches are in no sense complete, and so we make no claim towards classification in any particular case. Currently, BLT-sets have been classified for q ≤ 17, by various authors. Thas (1987) proved that for q = 3 there arises only the linear flock, and hence only the classical BLT-set. For q = 5 and q = 7, De Clerck, Gevaert and Thas (1988) proved there is exactly one nonlinear flock, and so there is exactly one nonclassical BLT-set, equivalent to the example of Fisher. For q = 9, Mylle (1991) showed with the use of a computer that there exist exactly two nonlinear flocks, and so there exist two nonclassical BLT-sets, equivalent to the Fisher example and to the Kantor/Knuth semifield example. De Clerck and Herssens (1992) used a computer to show that there exist exactly three nonlinear flocks for q = 11, giving rise to the BLT-sets of Fisher, Fisher-Thas-Walker/Kantor and Penttila(DCHT (11)). Finally, Penttila and Royle (1998) classified BLT-sets using a computer for q = 13 and q = 17, showing that there exist exactly two and five nonclassical BLT-sets in each respective case. A future paper by Penttila and the present author classifies BLT-sets for 19 ≤ q ≤ 29 using a computer.

7.1 Search techniques

The techniques for searching for BLT-sets which we shall discuss here are different to those of the earlier searches mentioned above, and have been implemented using the computer algebra system MAGMA [32]. Here we discuss only some of the generic concepts relevant to these searches, which also have the potential to be used in different situations, that is, for searching for different structures. Note that a great deal of time can be spent in any particular case experimenting (or “playing around”) with the structures and the spaces they live in, determining the feasibility of various approaches.

The techniques used by Penttila and Royle (1998) involved computing the automorphism groups of the generalised quadrangles Q(4, q) for q ≤ 25, which we have seen are known to be P Γ O(5, q). In terms of computer time, this was by far the most expensive part of their work. As a result of recent upgrades, MAGMA now has P GO(5, q) for q ≤ 97 implemented in it as a permutation group acting on the points of the associated parabolic quadric Q(4, q). It is a trivial matter to extend these groups to the full groups P Γ O(5, q) necessary for the searches.

The theme of our searches is the use of nontrivial symmetry hypotheses. This involves the choice of a subgroup H of P Γ O(5, q) and the determination of all BLTsets that are H-invariant. Since we are only interested in BLT-sets up to equivalence, we are only interested in conjugacy classes of subgroups. Such a search determines, up to equivalence, all BLT-sets with stabiliser containing a conjugate of H. The weakest nontrivial symmetry hypothesis is to use a minimal subgroup of P Γ O(5, q). The minimal subgroups of a finite group - the atoms of the lattice of subgroups - are the subgroups of prime order. So our basic philosophy is to loop through the primes p dividing |P Γ O(5, q)| in descending order of magnitude, take one representative H of each conjugacy class of subgroups of P Γ O(5, q) of order p, and determine all BLT-sets that are H-invariant. This “prime-at-a-time” technique has previously been used for hyperovals by O’Keefe and Penttila (1992) and Penttila and Royle (1995).

The determination of all BLT-sets admitting a subgroup H is achieved by enlarging BLT-sets to the hereditary class of partial BLT-sets and then stitching together Horbits. Namely a graph with vertices H-orbits that are partial BLT-sets and edges pairs of vertices with union a partial BLT-set is (perhaps only conceptually) formed. Every BLT-set is a clique of this graph with union of the vertices containing q + 1 points of Q(4, q), although not every such clique need be a BLT-set.

Finally, using the results on discriminants of Bader, O’Keefe and Penttila (submitted) in Section 6.1, we have available a computationally inexpensive and easily implemented check for determining if a set of points of Q(4, q) is a partial BLT-set. The significance of these results is that we may test a set of points for being a partial BLT-set by performing a simple field calculation on all triples in the set containing a chosen element of the set, instead of having to test collinearity of all points of Q(4, q) with each point of the set. This plays a major role in making these searches successful.

As mentioned above, our searches are not considered to be complete in any particular case. For, although in theory we know how to determine up to equivalence all BLT-sets with nontrivial stabiliser, in practice this is sometimes not feasible. Thus we do not always determine the BLT-sets up to equivalence, nor determine all conjugacy classes of subgroups to work with. Instead, we often only work up to certain invariants, such as the order of the stabiliser or number of fixed points of elements of the subgroup, which we find appealing or are inexpensive to compute. Furthermore, small primes, particularly 2 and 3, are often beyond our means, and so a selection of subgroups properly containing such subgroups is chosen as an alternative. Note however, that by Sylow’s theorem, the conjugacy classes of subgroups of prime order p are all represented within a single Sylow p-subgroup.

In the following, the code fragments presented are by no means the most efficient, but merely for instructive purposes as examples of the MAGMA syntax.

When we call the projective general orthogonal group P GO(5, q) of five dimensions over the Galois field of order q = p^{h}, MAGMA indexes the points of the parabolic quadric Q(4, q) and computes P GO(5, q) as a permutation group acting on this set of indices. We label the field, group and indexed set I as follows :

> q:=p^h;
> F:=GaloisField(q);
> G,I:=ProjectiveGeneralOrthogonalGroup(5,F);

We remark that I is a sequence of q^{3} +q^{2} +q +1 points of PG(4, q), each represented by a (homogeneous) 5-tuple from GF (q), normalised so that the leftmost nonzero coordinate is a 1.

The quadratic form used by MAGMA to represent Q(4, q) is

Q(x) = x_{1}x_{5} + x_{2}x_{4} + \frac{1}{4}x_{3}^{2}

(up to a scalar multiple). This can be verified by the following code :

> Q:=func<x|F!(x[1]*x[5]+x[2]*x[4]+x[3]^2/4)>;
> for i:=1 to #I do
>  if Q(I[i]) ne 0 then print i;end if;
> end for;

and observing no output occurring. (This does not seem to be documented in the manual for MAGMA.)

The full group of the quadric Q(4, q) is the projective semilinear orthogonal group P Γ O(5, q), which for prime order fields is equivalent to P GO(5, q). In general, we can construct P Γ O(5, q) by forming the subgroup of the symmetric group on the set of indices generated by P GO(5, q) and a field automorphism, namely, the Frobenius automorphism x ↦ x^{p}.

> V:=VectorSpace(F,5);
> a:=[];
> for i:=1 to #I do x:=I[i];
>  Append(~a,Index(I,V![x[1]^p,x[2]^p,x[3]^p,x[4]^p,x[5]^p]));
> end for;
> PGammaO:=sub<Sym(#I)|G,a>;

To determine if a set of points of Q(4, q) is a partial BLT-set, we implement the results on discriminants of Bader, O’Keefe and Penttila (submitted) in Section 6.1. This can be done as follows :

> Q:=func<x|x[1]*x[5]+x[2]*x[4]+x[3]^2/4>;
> f:=func<x,y|Q(x+y)-Q(x)-Q(y)>;
> d:=((q-1) div 2);
> blt:=function(X);
>  if #X ge 3 then x:=Random(X);
>   for y in X diff {x} do flag:=true;
>    for z in X diff {x,y} do
>     flag:=(-4*f(I[x],I[y])*f(I[x],I[z])*f(I[y],I[z]))^d eq F!(-1);
>     if not flag then break;end if;
>    end for;
>    if not flag then break;end if;
>   end for;
>   return flag;
>  else return true;
>  end if;
> end function;

Our basic pattern of searching is for each prime p dividing |P Γ O(5, q)| to compute a Sylow p-subgroup of P Γ O(5, q)

> P:=Sylow(PGammaO,p);

loop over the elements of P of order p

> list:={};
> for g in P do
>  if Order(g) eq p then
>   H:=sub<P|g>; FP:=Fix(H);
>   if not done(H,list) then
>     O:=Orbits(H);
>     cand:={i:i in {1..#O}|blt(O[i])};
>     c:={i:i in cand|#O[i] eq p};
>     edges:={{i,j}:i in c,j in c|i lt j and blt(O[i] join O[j])};
>     Gamma:=Graph<c|edges>;
>     d:=((q+1-#FP) div p)+1;e:=(q+1) div p;
>     for k:=d to e do
>      Cl:=AllCliques(Gamma,k);
>      for j:=1 to #Cl do Y:={};
>       for k in {x:x in c|VertexSet(Gamma)!x in Cl[j]} do
>        Y:=Y join O[k];
>       end for;
>       for X in Subsets(FP,(q+1)-k*p) do
>        if blt(Y join X) then
>         PrintFile(``bltsets'',Y join X);list:=list join {H};
>        end if;
>       end for;
>      end for;
>     end for;
>   end if;
>  end if;
> end for;

and provided that the subgroup H it generates has not already been dealt with up to conjugacy, calculate and write all H-invariant BLT-sets to a file. Here the function for testing whether a subgroup has already been dealt with can be implemented as follows :

> done:=function(H,list);
>  flag:=false;
>  for X in list do flag:=IsConjugate(PGammaO,H,X);
>   if flag then break;end if;
>  end for;
>  return flag;
> end function;

In the interest of efficiency, not all of these computations need be performed. For instance, conjugacy testing need only be performed if (cheap) invariants indicate the subgroups may be conjugate, and normalisers may be used to not have to calculate all H-invariant BLT-sets, reducing, in the process, the size of the cliques that need to be computed.

Finally, we must have some way of determining whether a BLT-set that is discovered is in fact new. For most of the searching done the order of the stabiliser of the BLTset in P Γ O(5, q) was used.

> GB:=Stabiliser(PGammaO,B);

This proved to be a reasonably effective test in most cases, though it should be noted that it was not always sufficient, and stricter testing was sometimes required; for example, comparing the order and number of fixed points of elements of the stabilisers. Indeed, this is how the new example for q = 37 with group order 72 was discovered to be different from the Kantor monomial example, which also has order 72 for q = 37. This highlights the fact that our searches have been far from exhaustive, and that possibly some new examples were overlooked during searching because they had the same group orders as known examples. Other more complex invariants could be used, including an adaptation of the conic invariant of Williams (1998). However, these simple invariants are enough to distinguish the examples presented in the following chapter.

98  Chapter 7. Searching for BLT-sets

CHAPTER 8 The new BLT-sets

In this chapter we present the new examples arising from the searches discussed in the previous chapter, along with some new examples discovered by other ad hoc searches. The results of the searches are summarised in the following table, along with the known examples not yet belonging to an infinite family. The group order listed is that of the stabiliser of the BLT-set in P Γ O(5, q), and the orbit structure is listed as a multiset, so, e.g., {4^{2}, 8^{3}} indicates that there are two orbits of length 4 and three of length 8. As we saw earlier, the number of orbits of the stabiliser of a BLT-set in P Γ O(5, q) on the BLT-set is equal to the number of nonisomorphic flocks arising from the BLT-set, and so we list this number explicitly in the final column of the table.

We abbreviate the attributions discussed in the previous chapter as follows :

DCH - De Clerck and Herssens (1992) DCP - De Clerck and Penttila (1997) PR - Penttila and Royle (1998) PW/BLP - Penttila and Williams (2000) / Bader, Lunardon and Pinneri (1999)

For q = 27 we find one new example, for q = 29 five new examples, for q = 31 five new examples, for q = 37 three new examples, for q = 41 three new examples, for q = 43 one new example, for q = 47 two new examples, for q = 49 two new examples, for q = 53 two new examples, for q = 59 three new examples, for q = 71 one new example, for q = 83 one new example, for q = 89 one new example, and for q = 125 one new example, all of which will be presented in the following sections.

100  Chapter 8. The new BLT-sets

q Attribution Group Order Orbit Structure # flocks
17 DCH  144  {12,6}  2
PR  24  {12,6}  2
19 PR  20  {20}  1
PR  16  {8^{2}, 2^{2}}  4
23 DCH  72  {18,6}  2
PR  1152  {24}  1
PR  24  {24}  1
PR  16  {8^{2}, 4^{2}}  4
PR  6  {6^{3}, 3^{2}}  5
25 PR  16  {16,8,2}  3
PR  8  {8^{3}, 2}  4
27 New  6  {1^{2}, 2, 6^{4}}  7
29 New  720  {30}  1
New  48  {24,6}  2
New  8  {8^{3}, 4, 2}  5
New  6  {6^{4}, 3^{2}}  6
New  3  {3^{10}}  10
31 New  96  {24,6,2}  3
New  10  {10^{2}, 5^{2}, 2}  5
New  8  {8^{3}, 4^{2}}  5
New  4  {4^{8}}  8
New  4  {4^{7}, 2^{2}}  9
37 New  72  {36,2}  2
New  4  {4^{9}, 2}  10
New  4  {4^{9}, 1^{2}}  11
41 New  60  {30,12}  2
New  24  {24,12,6}  3
New  8  {8^{5}, 2}  6
43 New  4  {4^{11}}  11
47 DCP  2304  {48}  1
New  24  {24^{2}}  2
New  3  {3^{16}}  16
49 New  40  {40,10}  2
New  20  {20^{2}, 5^{2}}  4
53 New  24  {24^{2}, 6}  3
New  12  {12^{4}, 6}  5
59 New  120  {60}  1
New  24  {24^{2}, 12}  3
New  5  {5^{12}}  12
71 New  24  {24^{3}}  3
83 New  24  {12, 24^{3}}  4
89 New  24  {6, 12, 24^{3}}  5
125 New  72  {72, 24^{2}, 6}  4
243 PW/BLP  26730  {243,1}  2

8.1. The new examples for q ≤ 89  101

8.1 The new examples for q ≤ 89

In this section we present the new BLT-sets discovered for q ≤ 89, along with the known BLT-sets not yet belonging to an infinite family. Each example is presented as a set of q +1 points of the nondegenerate quadric Q(4, q) defined by the quadratic equation

Q(x) = x_{1}x_{5} + x_{2}x_{4} + x_{3}^{2}

and containing ⟨(0, 0, 0, 0, 1)⟩ and ⟨(1, 0, 0, 0, 0)⟩. Note that this quadratic form is different to the form used by MAGMA (see Section 7.1) and to the form used in earlier theoretical discussions (see Section 6.3). We identify each BLT-set by the order of its stabiliser in P Γ O(5, q), with the abbreviations of attributions as above. The orbits of the stabiliser on the BLT-set are distinguished by vertical bars, reading across the table from left to right. The example X_{4}^{cyc} for q = 31 has a cyclic stabiliser, whilst X_{4}^{noncyc} has stabiliser isomorphic to the non-cyclic group of order 4. The example X_{4}^{2} for q = 37 has two fixed points on the BLT-set, whilst X_{4}^{0} has two fixed points not on the BLT-set.

DCH_{144} ( 1, 5, 15, 2, 3 )  ( 1, 9, 3, 5, 14 )  ( 1, 7, 12, 9, 14 )
q = 17 ( 1, 4, 2, 13, 12 )  ( 1, 14, 0, 1, 3 )  ( 1, 11, 13, 16, 12 ) |
( 1, 2, 14, 7, 11 )  ( 1, 0, 0, 0, 0 )  ( 1, 12, 3, 4, 11 )
( 1, 10, 14, 10, 10 ) ( 1, 6, 5, 14, 10 ) ( 1, 16, 16, 6, 5 )
( 1, 3, 10, 3, 10 )  ( 1, 13, 15, 8, 11 ) ( 1, 8, 1, 12, 5 )
( 1, 15, 12, 15, 5 ) ( 0, 0, 0, 0, 1 )  ( 1, 1, 1, 11, 5 )
P R_{24} ( 1, 0, 0, 0, 0 )  ( 1, 7, 1, 4, 5 )  ( 1, 12, 12, 3, 7 )
q = 17 ( 1, 2, 10, 8, 3 )  ( 1, 11, 3, 6, 10 ) ( 1, 5, 0, 1, 12 )  |
( 1, 13, 16, 10, 5 ) ( 1, 3, 15, 14, 5 ) ( 1, 10, 9, 13, 10 )
( 1, 8, 8, 12, 10 ) ( 1, 9, 3, 9, 12 ) ( 1, 15, 5, 16, 7 )
( 1, 4, 14, 7, 14 ) ( 1, 6, 13, 5, 5 ) ( 0, 0, 0, 0, 1 )
( 1, 14, 2, 15, 7 ) ( 1, 1, 0, 11, 6 ) ( 1, 16, 8, 2, 6 )
P R_{20} ( 1, 0, 0, 0, 0 )  ( 1, 1, 12, 1, 7 )  ( 1, 14, 11, 18, 7 )
q = 19 ( 1, 11, 4, 4, 16 ) ( 1, 6, 16, 7, 6 )  ( 1, 4, 18, 5, 17 )
( 1, 9, 17, 11, 11 ) ( 1, 5, 3, 10, 17 ) ( 1, 18, 2, 8, 4 )
( 1, 12, 16, 15, 1 ) ( 1, 17, 13, 13, 9 ) ( 1, 2, 3, 6, 17 )
( 1, 7, 8, 14, 9 ) ( 1, 16, 8, 9, 1 ) ( 1, 10, 5, 12, 7 )
( 1, 8, 8, 17, 9 )  ( 1, 15, 13, 16, 9 ) ( 1, 3, 8, 2, 6 )
( 0, 0, 0, 0, 1 )  ( 1, 13, 6, 3, 1 )

P R_{16} ( 1, 3, 5, 5, 17 )  ( 1, 6, 10, 8, 4 )  | ( 1, 17, 10, 11, 17 )
q = 19 ( 1, 18, 14, 3, 16 ) | ( 1, 14, 17, 14, 9 )  ( 1, 13, 4, 10, 6 )
( 1, 0, 0, 0, 0 )  ( 1, 9, 10, 6, 17 )  ( 1, 1, 9, 9, 5 )
( 1, 10, 5, 18, 4 )  ( 1, 11, 1, 1, 7 )  ( 1, 12, 9, 2, 9 )  |
( 1, 2, 12, 13, 1 )  ( 1, 16, 1, 4, 11 )  ( 1, 7, 15, 17, 17 )
( 0, 0, 0, 0, 1 )  ( 1, 4, 7, 15, 5 )  ( 1, 15, 16, 12, 1 )
( 1, 8, 12, 7, 9 )  ( 1, 5, 14, 16, 9 )

DCH_{72} ( 1, 9, 11, 17, 2 )  ( 1, 7, 3, 6, 18 )  ( 0, 0, 0, 0, 1 )
q = 23 ( 1, 16, 7, 4, 2 )  ( 1, 22, 14, 7, 18 ) ( 1, 21, 5, 2, 2 )  |
( 1, 0, 0, 0, 0 )  ( 1, 11, 8, 15, 1 ) ( 1, 3, 17, 18, 2 )
( 1, 10, 20, 22, 1 ) ( 1, 2, 10, 1, 13 ) ( 1, 14, 22, 8, 2 )
( 1, 20, 14, 16, 13 ) ( 1, 4, 13, 19, 8 ) ( 1, 1, 19, 3, 4 )
( 1, 18, 9, 21, 1 ) ( 1, 6, 4, 20, 2 ) ( 1, 17, 9, 10, 2 )
( 1, 13, 10, 9, 13 ) ( 1, 5, 1, 12, 8 )  ( 1, 15, 3, 13, 3 )
( 1, 8, 18, 5, 4 )  ( 1, 19, 20, 14, 1 ) ( 1, 12, 16, 11, 3 )

P R_{1152} ( 1, 0, 0, 0, 0 )  ( 1, 10, 12, 7, 16 ) ( 1, 19, 0, 12, 2 )
q = 23 ( 1, 17, 4, 13, 16 ) ( 1, 22, 13, 1, 16 ) ( 1, 6, 19, 10, 16 )
( 1, 13, 11, 16, 16 ) ( 1, 11, 19, 18, 16 ) ( 1, 4, 18, 2, 13 )
( 1, 1, 21, 11, 8 )  ( 1, 5, 10, 5, 13 )  ( 1, 14, 0, 4, 13 )
( 1, 15, 7, 15, 2 )  ( 1, 18, 5, 3, 13 )  ( 1, 3, 6, 21, 16 )
( 1, 20, 9, 17, 16 ) ( 0, 0, 0, 0, 1 )  ( 1, 16, 17, 14, 16 )
( 1, 8, 15, 9, 2 ) ( 1, 9, 2, 8, 16 ) ( 1, 21, 1, 20, 16 )
( 1, 7, 14, 19, 16 ) ( 1, 12, 8, 6, 2 )  ( 1, 2, 19, 22, 9 )

P R_{24} ( 1, 0, 0, 0, 0 )  ( 1, 3, 21, 11, 9 ) ( 1, 5, 7, 16, 9 )
q = 23 ( 1, 10, 0, 3, 16 )  ( 1, 7, 4, 17, 3 )  ( 1, 16, 16, 12, 12 )
( 1, 8, 10, 9, 12 )  ( 1, 17, 9, 6, 1 )  ( 1, 2, 5, 18, 8 )
( 1, 22, 14, 7, 18 ) ( 1, 1, 20, 8, 6 )  ( 1, 21, 17, 1, 12 )
( 1, 14, 19, 14, 18 ) ( 1, 9, 19, 2, 12 ) ( 1, 20, 6, 20, 1 )
( 1, 19, 21, 10, 13 ) ( 0, 0, 0, 0, 1 )  ( 1, 15, 3, 5, 8 )
( 1, 12, 12, 22, 6 ) ( 1, 6, 17, 15, 12 ) ( 1, 13, 16, 13, 12 )
( 1, 11, 16, 19, 18 ) ( 1, 4, 6, 21, 18 ) ( 1, 18, 18, 4, 18 )

P R_{16} ( 1, 0, 0, 0, 0 )  ( 0, 0, 0, 0, 1 )  ( 1, 11, 1, 18, 8 )
q = 23 ( 1, 7, 7, 6, 1 )  | ( 1, 16, 21, 16, 16 )  ( 1, 21, 16, 19, 12 )
( 1, 5, 0, 2, 13 )  ( 1, 18, 0, 21, 13 ) | ( 1, 20, 2, 17, 1 )
( 1, 13, 4, 7, 8 )  ( 1, 10, 15, 15, 16 )  ( 1, 17, 18, 3, 16 )
( 1, 12, 20, 12, 8 )  ( 1, 1, 18, 9, 12 )  ( 1, 14, 17, 13, 12 )
( 1, 4, 5, 5, 1 )  | ( 1, 22, 21, 8, 4 )  ( 1, 19, 18, 1, 2 )
( 1, 6, 19, 20, 2 )  ( 1, 8, 6, 4, 1 )  ( 1, 2, 15, 10, 8 )
( 1, 3, 3, 11, 4 )  ( 1, 9, 13, 22, 1 )  ( 1, 15, 14, 14, 8 )

P R_{6} ( 1, 8, 0, 22, 8 ) ( 1, 5, 15, 9, 6 )  ( 1, 11, 18, 13, 16 ) |
q = 23 ( 1, 17, 19, 7, 3 ) ( 1, 1, 4, 12, 18 ) ( 1, 4, 10, 19, 8 )  |
( 1, 0, 0, 0, 0 )  ( 1, 21, 8, 15, 12 ) ( 1, 13, 12, 10, 2 )
( 1, 15, 5, 8, 16 ) ( 1, 2, 16, 20, 3 ) ( 1, 6, 14, 18, 18 ) |
( 0, 0, 0, 0, 1 ) ( 1, 9, 9, 21, 6 ) ( 1, 22, 22, 2, 1 )
( 1, 10, 7, 5, 16 ) ( 1, 14, 15, 6, 13 ) ( 1, 19, 14, 16, 6 ) |
( 1, 18, 0, 11, 9 ) ( 1, 20, 8, 1, 8 )  ( 1, 7, 4, 17, 3 )
( 1, 12, 8, 4, 3 ) ( 1, 3, 18, 3, 12 ) ( 1, 16, 4, 14, 13 )

P R_{16}  (1, w^{5}, w^{21}, w^{17}, w^{5})  (1, w^{13}, w^{17}, 2, w^{9}) | (1, 2, w^{14}, w^{11}, w^{21})
q = 25  (1, w^{23}, w^{23}, w^{22}, w^{7})  (1, 4, w^{16}, 3, w^{11})  (1, w^{16}, 1, w^{7}, w^{9})
w^{2} = w − 2 (1, w^{3}, w^{10}, w^{16}, w^{5})  (1, w, w^{13}, w^{23}, w^{5})  (0, 0, 0, 0, 1)
(1, w^{15}, w^{8}, w^{15}, w^{7}) | (1, w^{8}, w^{8}, 4, w^{3})  (1, 0, 0, 0, 0)
(1, w^{10}, w^{13}, w^{5}, w^{19})  (1, w^{9}, w^{9}, w^{4}, w^{15})  (1, w^{4}, 2, w^{21}, w^{5})
(1, 1, w^{15}, w^{20}, w^{21})  (1, w^{7}, w^{5}, 1, w^{5})  (1, 3, w^{7}, w^{19}, w^{23})
(1, w^{2}, w^{15}, w^{2}, w^{9})  (1, w^{11}, w^{21}, w^{3}, w)  (1, w^{20}, 3, w^{13}, w^{7})
(1, w^{22}, w^{14}, w, w)  (1, w^{21}, 3, w^{10}, w^{9})  (1, w^{17}, w^{9}, w^{14}, w^{11})
(1, w^{19}, w^{7}, w^{9}, w^{5})  (1, w^{14}, 1, w^{8}, w^{3})

P R_{8}  (1, 1, w^{8}, w^{9}, w^{5})  (1, w^{23}, w^{21}, w^{21}, w^{23}) | (1, 0, 0, 0, 0)
q = 25  (0, 0, 0, 0, 1)  (1, w^{8}, 3, w^{13}, w^{11})  (1, w^{20}, w^{4}, w^{14}, w^{13})
w^{2} = w − 2 (1, w^{10}, 3, w^{15}, w^{5})  (1, w^{3}, w^{10}, w^{16}, w^{5})  (1, w^{11}, w^{4}, 2, w^{7})
(1, 3, w^{2}, w^{19}, w^{3})  | (1, w^{9}, 0, w^{22}, w^{19})  (1, w^{4}, 1, 3, w^{3})
(1, w^{22}, w^{10}, w^{20}, w^{23})  (1, w^{16}, w^{21}, w^{4}, w^{23})  (1, w^{17}, w^{13}, w^{5}, w^{9})
(1, w^{15}, w^{4}, 4, w^{5})  (1, w^{19}, w^{23}, w, w)  (1, w^{7}, w^{5}, 1, w^{5})  |
(1, 2, w^{8}, w^{8}, w^{19})  (1, w^{14}, w^{5}, w^{10}, w)  (1, 4, w^{4}, w^{17}, w^{3})
(1, w^{13}, w^{21}, w^{3}, w^{21})  (1, w^{5}, 2, w^{11}, w^{23})  (1, w^{2}, 4, w^{7}, w^{23})
(1, w, w^{13}, w^{23}, w^{5})  (1, w^{21}, 4, w^{2}, w^{9})

X_{6}  (0, 0, 0, 0, 1)  | (1, 0, 0, 0, 0)  (1, w, w^{19}, 2, w^{20})  |
q = 27  (1, w^{14}, w^{6}, 1, w^{20})  | (1, w^{20}, w^{18}, w^{17}, w^{6})  (1, w^{25}, w^{24}, w^{25}, w^{4})
w^{3} = w − 1 (1, w^{3}, w^{6}, w^{24}, w^{24})  (1, w^{16}, w^{5}, w^{23}, w^{24})  (1, w^{15}, w^{7}, w^{8}, w^{4})
(1, w^{5}, w^{21}, w^{20}, w^{6}) | (1, w^{24}, w^{5}, w^{14}, w^{18})  (1, w^{7}, w^{2}, w^{16}, w^{14})
(1, 2, w^{12}, w^{11}, w^{24}) (1, w^{23}, w^{6}, w^{7}, w^{6}) (1, w^{17}, w^{3}, w^{21}, w^{4})
(1, w^{11}, w^{16}, w^{22}, w^{2}) | (1, w^{8}, w^{24}, w^{15}, w^{18})  (1, w^{21}, 0, w^{12}, w^{20})
(1, w^{18}, w^{14}, w^{4}, w^{20})  (1, 1, w^{22}, w^{19}, w^{14})  (1, w^{2}, w^{7}, w^{10}, w^{20})
(1, w^{6}, w^{9}, w^{5}, w^{2})  | (1, w^{4}, w^{8}, w^{6}, w^{8})  (1, w^{19}, w^{23}, w^{2}, w^{16})
(1, w^{10}, w^{5}, w^{9}, 1)  (1, w^{9}, w^{22}, w, w^{12})  (1, w^{12}, w^{14}, w^{18}, w^{10})

(1, w^{22}, w^{5}, w^{3}, w^{22})

X_{720} ( 1, 26, 16, 3, 14 ) ( 1, 0, 0, 0, 0 )  ( 1, 20, 11, 4, 2 )
q = 29 ( 1, 6, 27, 27, 8 ) ( 1, 16, 20, 2, 3 ) ( 1, 19, 15, 9, 10 )
( 1, 10, 19, 28, 26 ) ( 0, 0, 0, 0, 1 )  ( 1, 9, 28, 24, 15 )
( 1, 13, 9, 7, 2 )  ( 1, 14, 11, 15, 17 ) ( 1, 27, 24, 23, 21 )
( 1, 23, 22, 16, 18 ) ( 1, 7, 17, 6, 17 )  ( 1, 15, 10, 25, 18 )
( 1, 5, 20, 26, 21 ) ( 1, 24, 26, 11, 17 ) ( 1, 22, 0, 17, 3 )
( 1, 3, 8, 5, 8 )  ( 1, 8, 25, 21, 19 ) ( 1, 2, 24, 8, 17 )
( 1, 1, 0, 12, 17 )  ( 1, 25, 27, 22, 26 ) ( 1, 18, 8, 18, 18 )
( 1, 21, 0, 20, 15 ) ( 1, 12, 24, 14, 10 ) ( 1, 17, 23, 19, 18 )
( 1, 11, 0, 13, 2 )  ( 1, 28, 17, 1, 2 )  ( 1, 4, 4, 10, 2 )

X_{48} ( 1, 26, 24, 25, 21 ) ( 1, 15, 3, 24, 8 )  ( 1, 5, 14, 3, 21 )
q = 29 ( 1, 18, 23, 20, 10 ) ( 1, 3, 28, 13, 18 ) ( 1, 14, 26, 5, 8 )  |
( 0, 0, 0, 0, 1 ) ( 1, 0, 0, 0, 0 ) ( 1, 2, 23, 17, 17 )
( 1, 12, 17, 18, 17 ) ( 1, 20, 19, 7, 21 ) ( 1, 21, 12, 27, 14 )
( 1, 7, 28, 1, 21 )  ( 1, 24, 8, 28, 18 ) ( 1, 19, 19, 15, 21 )
( 1, 11, 25, 21, 14 ) ( 1, 25, 11, 2, 3 )  ( 1, 13, 7, 8, 21 )
( 1, 17, 18, 26, 17 ) ( 1, 28, 19, 23, 10 ) ( 1, 6, 18, 6, 17 )
( 1, 23, 11, 11, 3 ) ( 1, 4, 12, 4, 14 )  ( 1, 8, 21, 12, 14 )
( 1, 9, 11, 10, 21 ) ( 1, 27, 9, 22, 21 ) ( 1, 10, 18, 9, 21 )
( 1, 22, 2, 16, 21 ) ( 1, 16, 19, 19, 2 ) ( 1, 1, 19, 14, 2 )

X_{8}  ( 1, 18, 7, 11, 14 )  ( 1, 15, 20, 28, 21 ) | ( 1, 13, 20, 8, 18 )
q = 29 ( 1, 24, 23, 24, 26 )  ( 1, 22, 26, 1, 27 )  ( 1, 7, 6, 13, 18 )  |
( 1, 0, 0, 0, 0 )  ( 1, 16, 11, 26, 14 )  ( 1, 27, 18, 12, 19 )
( 1, 14, 23, 20, 3 )  ( 1, 6, 5, 22, 17 )  ( 1, 11, 10, 6, 8 )
( 1, 26, 3, 2, 26 )  ( 1, 28, 22, 23, 3 ) | ( 1, 25, 24, 25, 17 )
( 1, 1, 22, 27, 11 )  ( 1, 2, 28, 5, 18 )  ( 1, 21, 13, 4, 8 )
( 1, 5, 24, 14, 21 )  ( 1, 10, 24, 16, 18 )  ( 0, 0, 0, 0, 1 )
( 1, 8, 25, 3, 18 )  | ( 1, 23, 15, 15, 10 )  ( 1, 4, 10, 21, 19 )
( 1, 3, 3, 10, 19 )  ( 1, 12, 16, 7, 8 )  ( 1, 9, 20, 18, 18 )
( 1, 17, 2, 17, 26 )  ( 1, 20, 27, 9, 19 )  ( 1, 19, 17, 19, 17 )

X_{6}  ( 1, 0, 0, 0, 0 )  ( 1, 9, 21, 11, 11 ) ( 1, 8, 16, 17, 14 ) |
q = 29 ( 1, 2, 20, 28, 8 ) ( 1, 1, 24, 1, 3 )  ( 1, 10, 7, 26, 10 ) |
( 1, 3, 1, 25, 11 ) ( 1, 20, 1, 21, 14 ) ( 1, 15, 17, 3, 14 )
( 1, 14, 0, 23, 26 ) ( 1, 21, 3, 19, 27 ) ( 0, 0, 0, 0, 1 )  |
( 1, 11, 28, 5, 2 ) ( 1, 24, 1, 24, 3 )  ( 1, 17, 4, 15, 19 )
( 1, 27, 11, 4, 3 ) ( 1, 4, 6, 10, 11 )  ( 1, 25, 26, 22, 21 ) |
( 1, 19, 18, 8, 17 ) ( 1, 26, 21, 7, 15 ) ( 1, 16, 7, 9, 10 )
( 1, 5, 3, 16, 27 ) ( 1, 22, 17, 14, 12 ) ( 1, 23, 27, 12, 10 ) |
( 1, 6, 16, 13, 14 ) ( 1, 12, 15, 27, 2 ) ( 1, 18, 5, 6, 12 )
( 1, 7, 26, 18, 10 ) ( 1, 13, 3, 20, 21 ) ( 1, 28, 4, 2, 15 )

X_{3}  ( 1, 0, 0, 0, 0 )  ( 1, 23, 28, 3, 17 ) ( 1, 22, 20, 12, 3 ) |
q = 29 ( 1, 16, 25, 7, 17 ) ( 1, 20, 27, 9, 19 ) ( 0, 0, 0, 0, 1 )  |
( 1, 2, 16, 18, 27 ) ( 1, 19, 4, 14, 8 ) ( 1, 8, 0, 25, 3 ) |
( 1, 25, 9, 16, 12 ) ( 1, 14, 20, 11, 26 ) ( 1, 9, 28, 26, 26 ) |
( 1, 6, 9, 15, 3 )  ( 1, 12, 6, 13, 11 ) ( 1, 27, 5, 4, 12 ) |
( 1, 21, 22, 1, 17 ) ( 1, 17, 16, 24, 3 ) ( 1, 1, 25, 2, 11 ) |
( 1, 13, 26, 8, 3 )  ( 1, 18, 13, 27, 12 ) ( 1, 11, 1, 20, 11 ) |
( 1, 26, 6, 19, 21 ) ( 1, 7, 9, 17, 3 )  ( 1, 4, 16, 5, 14 ) |
( 1, 15, 5, 28, 19 ) ( 1, 5, 5, 21, 15 )  ( 1, 28, 7, 10, 19 ) |
( 1, 24, 11, 22, 18 ) ( 1, 3, 9, 6, 17 )  ( 1, 10, 9, 23, 8 )

X_{96} ( 1, 0, 0, 0, 0 )  ( 0, 0, 0, 0, 1 )  | ( 1, 30, 30, 3, 2 )
q = 31 ( 1, 1, 1, 28, 2 )  ( 1, 6, 6, 13, 10 )  ( 1, 5, 5, 16, 19 )
( 1, 25, 25, 18, 10 ) ( 1, 26, 26, 15, 19 ) | ( 1, 12, 3, 1, 10 )
( 1, 21, 13, 25, 19 ) ( 1, 22, 10, 27, 19 )  ( 1, 4, 24, 14, 19 )
( 1, 3, 15, 22, 19 ) ( 1, 16, 24, 19, 19 )  ( 1, 28, 11, 10, 2 )
( 1, 23, 2, 24, 2 )  ( 1, 2, 13, 30, 19 )  ( 1, 11, 28, 20, 19 )
( 1, 19, 15, 6, 2 )  ( 1, 8, 16, 21, 10 )  ( 1, 29, 15, 5, 2 )
( 1, 27, 18, 4, 2 )  ( 1, 15, 13, 17, 10 )  ( 1, 18, 27, 2, 10 )
( 1, 7, 11, 9, 2 )  ( 1, 24, 16, 7, 10 )  ( 1, 14, 28, 29, 19 )
( 1, 17, 19, 11, 10 ) ( 1, 9, 18, 12, 2 )  ( 1, 10, 3, 26, 10 )
( 1, 20, 27, 8, 10 ) ( 1, 13, 3, 23, 2 )

X_{10} ( 1, 0, 0, 0, 0 )  ( 0, 0, 0, 0, 1 )  | ( 1, 30, 3, 18, 9 )
q = 31 ( 1, 14, 14, 20, 20 )  ( 1, 28, 17, 17, 10 )  ( 1, 12, 10, 16, 18 )
( 1, 9, 18, 22, 5 )  | ( 1, 15, 8, 14, 5 )  ( 1, 2, 1, 6, 18 )
( 1, 11, 5, 11, 9 )  ( 1, 24, 18, 30, 10 )  ( 1, 10, 30, 1, 20 ) |
( 1, 4, 18, 7, 20 )  ( 1, 13, 4, 10, 9 )  ( 1, 23, 16, 21, 5 )
( 1, 8, 18, 19, 20 )  ( 1, 18, 24, 29, 18 )  ( 1, 3, 2, 3, 18 )
( 1, 25, 2, 28, 9 )  ( 1, 19, 2, 27, 10 )  ( 1, 26, 6, 2, 5 )
( 1, 16, 1, 9, 10 )  | ( 1, 5, 20, 25, 2 )  ( 1, 7, 28, 4, 25 )
( 1, 21, 9, 24, 4 )  ( 1, 6, 13, 13, 1 )  ( 1, 20, 14, 5, 14 )
( 1, 17, 25, 15, 19 )  ( 1, 1, 10, 8, 16 )  ( 1, 29, 10, 23, 8 )
( 1, 22, 14, 26, 7 )  ( 1, 27, 12, 12, 28 )

X_{8}  ( 1, 0, 0, 0, 0 )  ( 1, 29, 27, 17, 18 )  ( 1, 5, 22, 20, 5 )
q = 31 ( 1, 26, 8, 6, 28 )  | ( 1, 8, 20, 2, 18 )  ( 1, 27, 15, 10, 1 )
( 1, 21, 0, 4, 9 )  ( 1, 9, 22, 15, 1 )  | ( 1, 22, 7, 30, 4 )
( 1, 28, 14, 5, 5 )  ( 1, 4, 24, 7, 16 )  ( 1, 10, 3, 19, 18 )
( 0, 0, 0, 0, 1 )  ( 1, 17, 23, 14, 8 )  ( 1, 30, 26, 8, 14 )
( 1, 11, 12, 11, 14 ) | ( 1, 18, 6, 27, 5 )  ( 1, 24, 14, 13, 19 )
( 1, 6, 24, 29, 25 )  ( 1, 2, 23, 25, 10 )  ( 1, 13, 3, 23, 2 )
( 1, 20, 28, 1, 2 )  ( 1, 12, 1, 12, 10 )  ( 1, 1, 26, 21, 16 ) |
( 1, 3, 17, 26, 5 )  ( 1, 25, 5, 16, 9 )  ( 1, 23, 6, 9, 5 )
( 1, 14, 7, 18, 9 )  ( 1, 19, 12, 3, 16 )  ( 1, 16, 21, 28, 10 )
( 1, 15, 14, 22, 1 )  ( 1, 7, 4, 24, 2 )

X_{4}^{cyc} ( 1, 16, 1, 13, 8 )  ( 1, 29, 12, 19, 18 )  ( 1, 0, 0, 0, 0 )
q = 31 ( 1, 14, 4, 7, 10 )  | ( 1, 28, 15, 11, 25 )  ( 1, 27, 27, 1, 19 )
( 1, 19, 11, 18, 2 )  ( 1, 20, 4, 27, 2 )  | ( 1, 5, 1, 5, 5 )
( 1, 12, 10, 4, 7 )  ( 1, 25, 12, 6, 16 )  ( 1, 1, 26, 2, 4 )  |
( 1, 6, 16, 3, 5 )  ( 1, 24, 5, 9, 7 )  ( 1, 8, 4, 20, 10 )
( 1, 4, 14, 24, 18 ) | ( 0, 0, 0, 0, 1 )  ( 1, 9, 15, 29, 10 )
( 1, 3, 25, 23, 19 )  ( 1, 7, 11, 8, 9 )  | ( 1, 18, 23, 14, 25 )
( 1, 17, 18, 15, 10 )  ( 1, 22, 18, 21, 20 )  ( 1, 11, 25, 16, 5 ) |
( 1, 30, 10, 26, 19 )  ( 1, 26, 1, 30, 25 )  ( 1, 21, 26, 25, 8 )
( 1, 2, 13, 17, 14 ) | ( 1, 13, 4, 12, 14 )  ( 1, 15, 1, 22, 10 )
( 1, 23, 0, 10, 18 )  ( 1, 10, 20, 28, 2 )

X_{4}^{noncyc} ( 1, 14, 29, 2, 2 )  ( 1, 28, 26, 18, 9 ) | ( 1, 7, 21, 19, 28 )
q = 31 ( 1, 5, 13, 3, 28 ) | ( 1, 15, 27, 12, 2 )  ( 1, 30, 24, 29, 9 )
( 1, 23, 5, 30, 9 )  ( 1, 0, 0, 0, 0 )  | ( 0, 0, 0, 0, 1 )
( 1, 18, 26, 6, 2 )  ( 1, 4, 1, 28, 4 )  ( 1, 21, 12, 10, 2 ) |
( 1, 9, 22, 7, 2 )  ( 1, 10, 28, 11, 4 )  ( 1, 27, 23, 1, 19 )
( 1, 20, 15, 9, 4 ) | ( 1, 22, 25, 15, 2 )  ( 1, 26, 1, 16, 10 )
( 1, 25, 21, 25, 1 )  ( 1, 11, 13, 5, 19 ) | ( 1, 2, 0, 21, 20 )
( 1, 1, 15, 27, 2 )  ( 1, 3, 14, 26, 28 )  ( 1, 16, 3, 22, 10 ) |
( 1, 8, 8, 13, 4 )  ( 1, 29, 1, 24, 9 )  ( 1, 6, 15, 17, 20 )
( 1, 13, 8, 4, 25 ) | ( 1, 24, 0, 8, 25 )  ( 1, 12, 25, 23, 25 )
( 1, 19, 8, 14, 28 )  ( 1, 17, 5, 20, 18 )

X_{72} ( 0, 0, 0, 0, 1 )  ( 1, 0, 0, 0, 0 )  | ( 1, 20, 1, 33, 5 )
q = 37 (1, 6, 23, 19, 23 )  ( 1, 32, 11, 35, 17 )  ( 1, 33, 12, 14, 23 )
( 1, 25, 8, 12, 6 )  ( 1, 16, 6, 1, 22 )  ( 1, 23, 8, 5, 6 )
( 1, 8, 6, 13, 8 )  ( 1, 22, 19, 20, 13 )  ( 1, 21, 23, 16, 23 )
( 1, 28, 6, 9, 8 )  ( 1, 2, 26, 7, 13 )  ( 1, 5, 9, 36, 35 )
( 1, 12, 23, 10, 17 ) ( 1, 14, 16, 25, 23 )  ( 1, 9, 8, 26, 35 )
( 1, 1, 11, 8, 19 ) ( 1, 19, 12, 11, 17 ) ( 1, 18, 27, 22, 22 )
( 1, 24, 36, 17, 35 ) ( 1, 26, 27, 23, 5 )  ( 1, 27, 25, 24, 22 )
( 1, 36, 21, 32, 35 ) ( 1, 7, 16, 31, 8 )  ( 1, 35, 5, 15, 5 )
( 1, 10, 36, 6, 13 ) ( 1, 30, 4, 3, 5 )  ( 1, 4, 3, 30, 19 )
( 1, 17, 13, 2, 19 ) ( 1, 11, 28, 18, 17 )  ( 1, 13, 16, 27, 22 )
( 1, 29, 12, 28, 6 ) ( 1, 15, 10, 34, 19 )  ( 1, 3, 30, 4, 13 )
( 1, 31, 9, 21, 8 )  ( 1, 34, 9, 29, 6 )

X_{4}^{0} ( 1, 9, 16, 8, 5 )  ( 1, 13, 20, 26, 2 ) | ( 1, 0, 0, 0, 0 )
q = 37 ( 1, 4, 12, 14, 22 )  ( 1, 19, 23, 5, 5 )  ( 1, 24, 17, 19, 32 ) |
( 0, 0, 0, 0, 1 )  ( 1, 18, 18, 10, 14 )  ( 1, 2, 4, 36, 23 )
( 1, 7, 5, 24, 29 )  | ( 1, 36, 20, 22, 29 )  ( 1, 14, 10, 13, 14 )
( 1, 28, 22, 7, 23 )  ( 1, 32, 28, 20, 19 ) | ( 1, 30, 27, 23, 24 )
( 1, 3, 30, 28, 15 )  ( 1, 1, 20, 31, 13 )  ( 1, 11, 36, 33, 6 ) |
( 1, 29, 22, 30, 15 )  ( 1, 20, 10, 4, 5 )  ( 1, 8, 14, 32, 29 )
( 1, 25, 15, 9, 31 ) | ( 1, 26, 13, 18, 29 )  ( 1, 23, 19, 6, 19 )
( 1, 35, 11, 15, 20 )  ( 1, 22, 11, 25, 32 ) | ( 1, 10, 16, 2, 20 )
( 1, 17, 18, 27, 31 )  ( 1, 15, 29, 16, 29 )  ( 1, 31, 18, 1, 15 ) |
( 1, 12, 34, 35, 15 )  ( 1, 33, 23, 17, 20 )  ( 1, 27, 23, 21, 14 )
( 1, 34, 22, 3, 6 )  | ( 1, 21, 11, 11, 18 )  ( 1, 5, 33, 12, 35 )
( 1, 6, 22, 34, 15 )  ( 1, 16, 31, 29, 18 )

X_{4}^{2} ( 1, 3, 7, 25, 24 )  | ( 1, 35, 30, 32, 15 ) | ( 1, 0, 0, 0, 0 )
q = 37 ( 1, 4, 10, 30, 2 )  ( 1, 14, 8, 17, 31 )  ( 1, 30, 18, 2, 23 ) |
( 0, 0, 0, 0, 1 )  ( 1, 19, 0, 10, 32 )  ( 1, 8, 18, 12, 24 )
( 1, 34, 2, 20, 19 ) | ( 1, 28, 34, 15, 15 )  ( 1, 2, 21, 23, 31 )
( 1, 13, 29, 36, 23 )  ( 1, 23, 13, 8, 17 ) | ( 1, 25, 12, 35, 17 )
( 1, 1, 14, 4, 22 )  ( 1, 17, 2, 19, 6 )  ( 1, 12, 17, 1, 32 ) |
( 1, 27, 23, 3, 19 )  ( 1, 20, 2, 21, 20 )  ( 1, 33, 30, 22, 2 )
( 1, 11, 16, 31, 32 ) | ( 1, 29, 6, 13, 31 )  ( 1, 16, 17, 7, 6 )
( 1, 6, 25, 14, 31 )  ( 1, 32, 27, 29, 8 ) | ( 1, 15, 8, 9, 23 )
( 1, 5, 17, 24, 35 )  ( 1, 18, 13, 33, 14 )  ( 1, 7, 36, 26, 2 )  |
( 1, 21, 34, 28, 32 )  ( 1, 10, 22, 16, 22 )  ( 1, 26, 34, 27, 29 )
( 1, 31, 13, 34, 35 ) | ( 1, 9, 13, 18, 2 )  ( 1, 22, 28, 5, 31 )
( 1, 24, 3, 11, 23 )  ( 1, 36, 27, 6, 17 )

X_{60} ( 1, 27, 36, 26, 11 ) ( 1, 0, 0, 0, 0 )  ( 0, 0, 0, 0, 1 )
q = 41 ( 1, 22, 11, 6, 34 ) ( 1, 11, 40, 30, 38 ) ( 1, 24, 20, 33, 38 )
( 1, 38, 11, 3, 11 ) ( 1, 28, 1, 1, 12 )  ( 1, 35, 25, 31, 12 )
( 1, 12, 27, 36, 28 ) ( 1, 15, 31, 27, 28 ) ( 1, 34, 3, 12, 34 ) |
( 1, 32, 24, 37, 3 ) ( 1, 26, 1, 7, 22 )  ( 1, 30, 23, 2, 26 )
( 1, 4, 2, 34, 24 )  ( 1, 29, 29, 10, 17 ) ( 1, 8, 32, 25, 6 )
( 1, 16, 16, 22, 7 ) ( 1, 20, 1, 28, 13 ) ( 1, 36, 19, 29, 30 )
( 1, 18, 12, 20, 29 ) ( 1, 37, 10, 18, 13 ) ( 1, 14, 7, 35, 35 )
( 1, 33, 40, 14, 29 ) ( 1, 9, 21, 40, 19 ) ( 1, 40, 28, 32, 27 )
( 1, 23, 26, 16, 22 ) ( 1, 10, 20, 13, 3 ) ( 1, 17, 16, 39, 24 )
( 1, 13, 26, 38, 19 ) ( 1, 3, 13, 21, 14 ) ( 1, 5, 4, 8, 26 )
( 1, 21, 39, 5, 14 ) ( 1, 2, 39, 15, 7 )  ( 1, 1, 24, 9, 30 )
( 1, 19, 9, 23, 15 ) ( 1, 31, 14, 17, 15 ) ( 1, 25, 12, 24, 35 )
( 1, 7, 32, 11, 6 )  ( 1, 6, 8, 19, 27 )  ( 1, 39, 27, 4, 17 )

X_{24} ( 1, 25, 4, 20, 17 ) ( 1, 16, 12, 25, 30 ) ( 1, 22, 5, 30, 12 )
q = 41 ( 1, 15, 0, 8, 3 )  ( 1, 14, 3, 39, 19 ) ( 1, 26, 23, 2, 34 ) |
( 1, 3, 14, 40, 12 ) ( 1, 10, 2, 5, 28 )  ( 1, 5, 26, 23, 29 )
( 1, 37, 9, 34, 14 ) ( 1, 39, 40, 6, 11 ) ( 1, 13, 30, 33, 24 )
( 1, 11, 22, 18, 15 ) ( 1, 38, 34, 7, 13 ) ( 1, 28, 24, 29, 6 )
( 1, 23, 35, 17, 24 ) ( 1, 32, 29, 19, 27 ) ( 1, 33, 36, 4, 7 )  |
( 1, 21, 38, 36, 14 ) ( 1, 0, 0, 0, 0 )  ( 1, 7, 12, 13, 11 )
( 1, 20, 26, 28, 35 ) ( 1, 27, 36, 3, 17 ) ( 1, 24, 14, 16, 35 )
( 1, 40, 24, 9, 7 )  ( 1, 30, 17, 31, 11 ) ( 1, 36, 6, 37, 26 )
( 1, 4, 28, 21, 34 ) ( 1, 35, 23, 15, 12 ) ( 1, 1, 7, 14, 19 )
( 1, 18, 2, 35, 22 ) ( 1, 17, 30, 26, 11 ) ( 1, 31, 21, 27, 34 )
( 1, 34, 39, 22, 27 ) ( 1, 2, 13, 12, 12 ) ( 1, 9, 19, 24, 38 )
( 1, 12, 8, 1, 6 ) ( 1, 6, 25, 11, 6 ) ( 0, 0, 0, 0, 1 )
( 1, 29, 37, 38, 30 ) ( 1, 19, 40, 32, 6 ) ( 1, 8, 7, 10, 35 )

X_{8}  ( 1, 28, 16, 28, 26 )  ( 1, 5, 1, 25, 38 )  | ( 1, 8, 12, 30, 26 )
q = 41 ( 1, 0, 0, 0, 0 )  ( 1, 2, 27, 10, 30 )  ( 1, 10, 32, 18, 26 )
( 1, 36, 10, 32, 19 )  ( 1, 22, 8, 21, 7 )  ( 1, 9, 2, 9, 38 )
( 1, 35, 10, 26, 15 ) | ( 1, 6, 12, 33, 27 )  ( 0, 0, 0, 0, 1 )
( 1, 37, 29, 6, 3 )  ( 1, 24, 10, 34, 27 )  ( 1, 13, 24, 40, 11 )
( 1, 34, 15, 39, 7 )  ( 1, 17, 22, 17, 6 )  ( 1, 18, 13, 5, 28 ) |
( 1, 40, 31, 31, 13 )  ( 1, 12, 14, 3, 14 )  ( 1, 38, 40, 22, 24 )
( 1, 29, 6, 13, 38 )  ( 1, 31, 11, 27, 26 )  ( 1, 23, 12, 35, 35 )
( 1, 26, 15, 36, 28 )  ( 1, 30, 17, 29, 30 ) | ( 1, 39, 11, 14, 30 )
( 1, 33, 35, 12, 19 )  ( 1, 11, 24, 2, 17 )  ( 1, 21, 33, 24, 6 )
( 1, 27, 12, 20, 13 )  ( 1, 19, 16, 11, 27 )  ( 1, 25, 9, 16, 11 )
( 1, 1, 3, 15, 17 )  | ( 1, 16, 39, 4, 14 )  ( 1, 7, 36, 19, 6 )
( 1, 3, 11, 7, 22 )  ( 1, 4, 19, 8, 17 )  ( 1, 15, 14, 1, 35 )
( 1, 32, 21, 38, 24 )  ( 1, 20, 15, 37, 19 )  ( 1, 14, 10, 23, 29 )

\begin{aligned}X_{4} & ( 1, 13, 23, 21, 15 ) & & ( 1, 0, 0, 0, 0 ) & & ( 1, 27, 39, 11, 31 ) \\ q = 43 & ( 1, 1, 7, 40, 40 ) & | & ( 1, 39, 7, 4, 10 ) & & ( 1, 2, 6, 29, 35 ) \\ & ( 0, 0, 0, 0, 1 ) & & ( 1, 16, 8, 25, 9 ) & | & ( 1, 32, 16, 41, 23 ) \\ & ( 1, 10, 6, 1, 40 ) & & ( 1, 18, 37, 14, 13 ) & & ( 1, 12, 27, 18, 1 ) & | \\ & ( 1, 3, 6, 19, 36 ) & & ( 1, 20, 13, 36, 14 ) & & ( 1, 26, 31, 12, 17 ) \\ & ( 1, 11, 19, 37, 6 ) & | & ( 1, 40, 0, 26, 35 ) & & ( 1, 42, 36, 27, 21 ) \\ & ( 1, 25, 18, 2, 13 ) & & ( 1, 7, 39, 23, 38 ) & | & ( 1, 34, 28, 22, 16 ) \\ & ( 1, 19, 9, 7, 1 ) & & ( 1, 33, 2, 24, 21 ) & & ( 1, 14, 19, 10, 15 ) & | \\ & ( 1, 6, 21, 39, 13 ) & & ( 1, 31, 3, 8, 1 ) & & ( 1, 17, 35, 13, 16 ) \\ & ( 1, 41, 6, 38, 40 ) & | & ( 1, 21, 27, 30, 17 ) & & ( 1, 29, 14, 31, 23 ) \\ & ( 1, 5, 9, 42, 10 ) & & ( 1, 24, 6, 9, 6 ) & | & ( 1, 37, 35, 17, 38 ) \\ & ( 1, 36, 34, 15, 24 ) & & ( 1, 35, 10, 33, 35 ) & & ( 1, 38, 14, 34, 17 ) & | \\ & ( 1, 4, 23, 16, 9 ) & & ( 1, 8, 33, 20, 41 ) & & ( 1, 15, 24, 3, 24 ) \\ & ( 1, 30, 29, 5, 41 ) & | & ( 1, 28, 40, 28, 24 ) & & ( 1, 22, 2, 6, 36 ) \\ & ( 1, 23, 25, 32, 15 ) & & ( 1, 9, 31, 35, 14 ) \end{aligned} \begin{aligned}DCP_{2304} & ( 1, 12, 6, 19, 18 ) & ( 1, 0, 0, 0, 0 ) & ( 1, 23, 22, 36, 4 ) \\ q = 47 & ( 1, 25, 40, 13, 2 ) & ( 1, 13, 29, 5, 34 ) & ( 1, 44, 44, 30, 34 ) \\ & ( 1, 15, 43, 11, 7 ) & ( 1, 8, 41, 3, 34 ) & ( 1, 37, 7, 16, 17 ) \\ & ( 1, 16, 32, 22, 34 ) & ( 1, 41, 25, 34, 2 ) & ( 1, 27, 17, 17, 4 ) \\ & ( 1, 6, 11, 1, 14 ) & ( 1, 5, 35, 2, 34 ) & ( 1, 1, 23, 33, 2 ) \\ & ( 1, 31, 24, 31, 14 ) & ( 1, 10, 12, 27, 9 ) & ( 1, 38, 29, 45, 34 ) \\ & ( 1, 33, 12, 35, 17 ) & ( 1, 20, 33, 15, 21 ) & ( 1, 46, 38, 8, 21 ) \\ & ( 1, 42, 26, 6, 12 ) & ( 1, 21, 4, 12, 14 ) & ( 1, 36, 1, 9, 4 ) \\ & ( 1, 32, 6, 28, 8 ) & ( 1, 40, 43, 37, 8 ) & ( 1, 26, 11, 10, 42 ) \\ & ( 1, 11, 5, 38, 27 ) & ( 1, 34, 18, 42, 34 ) & ( 1, 7, 18, 4, 24 ) \\ & ( 0, 0, 0, 0, 1 ) & ( 1, 29, 5, 21, 24 ) & ( 1, 39, 25, 40, 24 ) \\ & ( 1, 3, 17, 25, 12 ) & ( 1, 17, 23, 29, 12 ) & ( 1, 2, 24, 14, 7 ) \\ & ( 1, 28, 23, 24, 21 ) & ( 1, 45, 33, 41, 27 ) & ( 1, 24, 28, 43, 17 ) \\ & ( 1, 18, 3, 46, 9 ) & ( 1, 19, 4, 18, 18 ) & ( 1, 22, 0, 7, 34 ) \\ & ( 1, 30, 38, 26, 32 ) & ( 1, 43, 6, 23, 9 ) & ( 1, 35, 36, 39, 18 ) \\ & ( 1, 9, 38, 32, 7 ) & ( 1, 14, 38, 44, 8 ) & ( 1, 4, 38, 20, 27 ) \end{aligned}

\begin{aligned}X_{24} & ( 1, 42, 19, 42, 37 ) & ( 1, 0, 0, 0, 0 ) & ( 1, 11, 4, 24, 2 ) \\ q = 47 & ( 1, 8, 30, 36, 34 ) & ( 1, 19, 42, 1, 3 ) & ( 1, 22, 36, 19, 25 ) \\ & ( 1, 28, 19, 7, 7 ) & ( 1, 16, 38, 12, 9 ) & ( 1, 34, 23, 11, 37 ) \\ & ( 1, 33, 25, 17, 36 ) & ( 1, 31, 23, 25, 12 ) & ( 1, 25, 3, 16, 14 ) \\ & ( 1, 23, 25, 30, 1 ) & ( 1, 13, 30, 3, 1 ) & ( 1, 35, 16, 40, 36 ) \\ & ( 1, 5, 4, 2, 21 ) & ( 1, 46, 35, 9, 6 ) & ( 1, 38, 3, 28, 8 ) \\ & ( 1, 43, 22, 5, 6 ) & ( 1, 29, 28, 35, 34 ) & ( 1, 9, 4, 33, 16 ) \\ & ( 1, 2, 24, 13, 9 ) & ( 1, 17, 14, 38, 4 ) & ( 1, 12, 23, 22, 6 ) & | \\ & ( 1, 44, 21, 15, 27 ) & ( 1, 39, 33, 18, 42 ) & ( 1, 18, 41, 44, 18 ) \\ & ( 1, 20, 38, 45, 6 ) & ( 1, 7, 24, 34, 32 ) & ( 1, 36, 23, 10, 4 ) \\ & ( 1, 24, 30, 27, 3 ) & ( 0, 0, 0, 0, 1 ) & ( 1, 45, 32, 32, 27 ) \\ & ( 1, 30, 43, 20, 42 ) & ( 1, 4, 18, 26, 42 ) & ( 1, 21, 32, 39, 37 ) \\ & ( 1, 37, 12, 43, 4 ) & ( 1, 26, 46, 23, 12 ) & ( 1, 14, 7, 31, 34 ) \\ & ( 1, 27, 11, 4, 6 ) & ( 1, 15, 17, 8, 14 ) & ( 1, 32, 17, 46, 25 ) \\ & ( 1, 41, 43, 41, 42 ) & ( 1, 6, 34, 21, 34 ) & ( 1, 1, 46, 29, 17 ) \\ & ( 1, 10, 11, 6, 7 ) & ( 1, 40, 25, 14, 37 ) & ( 1, 3, 34, 37, 2 ) \end{aligned} \begin{aligned}X_{3} & ( 1, 0, 0, 0, 0 ) & ( 1, 36, 44, 32, 14 ) & ( 1, 12, 12, 28, 37 ) & | \\ q = 47 & ( 1, 30, 43, 2, 18 ) & ( 1, 32, 37, 10, 3 ) & ( 0, 0, 0, 0, 1 ) & | \\ & ( 1, 24, 16, 40, 6 ) & ( 1, 43, 43, 18, 9 ) & ( 1, 18, 32, 44, 17 ) & | \\ & ( 1, 3, 38, 42, 28 ) & ( 1, 9, 35, 22, 34 ) & ( 1, 4, 23, 20, 2 ) & | \\ & ( 1, 17, 13, 37, 1 ) & ( 1, 44, 3, 35, 2 ) & ( 1, 5, 45, 25, 12 ) & | \\ & ( 1, 40, 37, 1, 1 ) & ( 1, 7, 43, 43, 12 ) & ( 1, 21, 35, 4, 7 ) & | \\ & ( 1, 39, 28, 16, 2 ) & ( 1, 1, 22, 6, 27 ) & ( 1, 26, 28, 8, 42 ) & | \\ & ( 1, 42, 43, 21, 42 ) & ( 1, 16, 4, 38, 34 ) & ( 1, 46, 43, 17, 1 ) & | \\ & ( 1, 45, 21, 26, 34 ) & ( 1, 33, 7, 36, 32 ) & ( 1, 13, 21, 13, 1 ) & | \\ & ( 1, 14, 43, 29, 1 ) & ( 1, 35, 33, 19, 32 ) & ( 1, 37, 29, 5, 8 ) & | \\ & ( 1, 25, 13, 23, 8 ) & ( 1, 11, 39, 11, 3 ) & ( 1, 19, 26, 46, 1 ) & | \\ & ( 1, 31, 3, 33, 2 ) & ( 1, 10, 8, 39, 16 ) & ( 1, 38, 25, 15, 27 ) & | \\ & ( 1, 27, 44, 27, 14 ) & ( 1, 15, 10, 34, 1 ) & ( 1, 22, 32, 30, 8 ) & | \\ & ( 1, 2, 14, 41, 4 ) & ( 1, 34, 17, 9, 16 ) & ( 1, 29, 12, 24, 6 ) & | \\ & ( 1, 6, 31, 3, 8 ) & ( 1, 8, 14, 14, 21 ) & ( 1, 41, 28, 45, 3 ) & | \\ & ( 1, 20, 9, 31, 4 ) & ( 1, 28, 36, 7, 12 ) & ( 1, 23, 40, 12, 4 ) \end{aligned}

\begin{aligned}X_{40} & (1, 0, 0, 0, 0) & & (1, w^{44}, w^{27}, 5, w^{33}) & (1, w^{6}, w^{27}, w^{3}, w^{27}) \\ q = 49 & (1, w^{21}, w^{11}, w^{7}, w^{45}) & & (1, w^{35}, w^{34}, 1, w^{33}) & (1, w^{26}, w^{33}, w^{17}, w) \\ w^{2} = w − 3 & (1, 5, 1, w^{10}, w^{35}) & & (1, w^{17}, 4, w, w^{3}) & (1, 4, 0, w^{33}, w^{41}) \\ & (1, w^{25}, 2, w^{25}, w^{29}) & | & (1, w^{27}, w^{26}, w^{45}, w^{23}) & (1, w^{20}, w^{43}, 2, w^{23}) \\ & (1, w^{14}, w^{25}, w^{43}, w^{13}) & & (1, 3, w^{37}, w^{22}, w^{19}) & (1, w^{41}, w^{7}, 6, w^{35}) \\ & (1, w^{30}, w^{34}, w^{11}, w^{23}) & & (1, w^{9}, 3, w^{14}, w^{27}) & (1, w^{15}, w^{41}, w^{5}, w^{25}) \\ & (1, w^{7}, w^{21}, w^{39}, w^{35}) & & (0, 0, 0, 0, 1) & (1, w^{23}, 6, w^{21}, w^{37}) \\ & (1, w, 6, w^{20}, w^{3}) & & (1, w^{12}, w^{41}, w^{26}, w^{27}) & (1, w^{18}, w^{22}, 4, w^{19}) \\ & (1, w^{19}, w^{27}, w^{44}, w) & & (1, w^{13}, 6, w^{37}, w^{35}) & (1, w^{10}, w^{19}, w^{9}, w^{39}) \\ & (1, w^{36}, 5, w^{46}, w^{19}) & & (1, w^{43}, w^{14}, w^{15}, w^{7}) & (1, w^{11}, w^{17}, w^{27}, w^{27}) \\ & (1, w^{33}, w^{5}, w^{34}, w^{5}) & & (1, w^{2}, w^{11}, w^{38}, w^{19}) & (1, w^{29}, w^{2}, w^{18}, w^{13}) \\ & (1, w^{37}, w^{29}, w^{35}, w^{15}) & & (1, 2, w^{38}, w^{41}, w^{29}) & (1, w^{47}, w^{7}, w^{13}, w^{47}) \\ & (1, w^{31}, w^{10}, w^{31}, w^{37}) & & (1, w^{28}, w^{17}, w^{4}, w^{19}) & (1, w^{3}, w^{29}, w^{36}, w^{11}) \\ & (1, w^{34}, w^{25}, w^{2}, w^{41}) & & (1, w^{22}, w^{44}, w^{47}, w^{41}) & (1, w^{4}, w^{7}, w^{12}, w) \\ & (1, w^{39}, w^{15}, w^{19}, w^{29}) & & (1, w^{45}, w^{39}, w^{6}, w^{33}) & (1, w^{46}, w^{20}, w^{28}, w^{31}) \\ & (1, w^{38}, 0, w^{29}, w^{43}) & & (1, 6, 4, w^{42}, w^{3}) & (1, w^{5}, w^{35}, w^{23}, w^{45}) \\ & (1, w^{42}, w^{19}, w^{30}, w^{29}) & & (1, 1, w^{25}, 3, w^{25}) \end{aligned} \begin{aligned}X_{20} & (1, 0, 0, 0, 0) & & (1, w^{21}, w^{2}, w^{27}, w^{41}) & & (1, w^{41}, 1, w^{20}, w^{15}) \\ q = 49 & (1, w, w^{12}, w^{37}, w^{29}) & & (1, w^{4}, 2, w^{43}, w^{45}) & | & (1, w^{14}, w^{5}, w^{2}, w^{33}) \\ w^{2} = w − 3 & (1, w^{28}, w^{46}, w^{19}, w^{17}) & & (1, w^{39}, w^{25}, 5, w^{3}) & & (1, w^{38}, w^{34}, 1, w^{17}) \\ & (1, 1, w^{39}, w^{13}, w^{7}) & | & (1, w^{7}, w^{22}, w^{46}, w^{39}) & & (1, w^{20}, w^{35}, w^{30}, w^{21}) \\ & (1, w^{9}, w^{18}, w^{9}, w^{15}) & & (1, 4, w^{31}, 6, w^{31}) & & (1, w^{10}, w, w^{6}, w^{7}) \\ & (1, w^{11}, w^{7}, w^{21}, w^{11}) & & (1, w^{29}, w^{45}, w^{28}, w^{7}) & & (1, w^{17}, w^{13}, w^{38}, w^{27}) \\ & (1, w^{37}, 3, w^{45}, w^{13}) & & (1, 5, w^{17}, w, w^{45}) & & (1, w^{6}, w^{13}, w^{35}, w^{39}) \\ & (1, w^{18}, w^{6}, w^{12}, w^{9}) & & (1, w^{2}, 6, w^{23}, w^{31}) & & (0, 0, 0, 0, 1) \\ & (1, w^{22}, w^{37}, w^{47}, w^{35}) & & (1, w^{44}, w^{33}, w^{31}, w^{13}) & & (1, w^{3}, w^{41}, 3, w^{17}) \\ & (1, 3, w^{29}, w^{5}, w^{31}) & & (1, w^{23}, w^{22}, w^{34}, w^{1}1) & & (1, w^{19}, w^{34}, w^{10}, w^{15}) & | \\ & (1, w^{5}, w^{35}, 4, w^{35}) & & (1, w^{43}, w^{33}, w^{29}, w^{41}) & & (1, w^{46}, w^{33}, w^{26}, w^{41}) \\ & (1, w^{42}, w^{20}, w^{42}, w^{29}) & & (1, w^{26}, w^{25}, w^{22}, w^{35}) & & (1, w^{13}, w^{35}, w^{7}, w^{7}) \\ & (1, w^{47}, w^{3}, w^{36}, w^{7}) & & (1, w^{45}, w^{4}, w^{15}, w) & & (1, w^{31}, w^{45}, w^{41}, w^{21}) \\ & (1, w^{25}, w^{38}, w^{18}, w^{41}) & & (1, w^{30}, w^{31}, w^{4}, w^{33}) & & (1, w^{27}, w^{30}, w^{14}, w^{13}) \\ & (1, w^{33}, 3, w^{33}, w^{3}) & & (1, 2, w^{25}, w^{17}, w^{27}) & & (1, w^{15}, w^{17}, w^{44}, w^{17}) \\ & (1, w^{36}, w^{33}, w^{39}, w^{13}) & & (1, 6, w^{19}, w^{25}, w^{45}) & & (1, w^{35}, w^{17}, w^{3}, w^{27}) \\ & (1, w^{12}, w^{45}, w^{11}, w^{43}) & & (1, w^{34}, w^{15}, 2, w) \end{aligned}

\begin{aligned}X_{24} & ( 1, 4, 23, 6, 30 ) & ( 1, 20, 37, 26, 19 ) & ( 1, 47, 33, 37, 34 ) \\ q = 53 & ( 1, 2, 40, 47, 2 ) & ( 1, 29, 10, 21, 33 ) & ( 1, 11, 32, 14, 41 ) & | \\ & ( 1, 0, 0, 0, 0 ) & ( 1, 24, 7, 35, 12 ) & ( 1, 26, 34, 16, 18 ) \\ & ( 1, 17, 51, 20, 27 ) & ( 1, 8, 11, 9, 19 ) & ( 1, 45, 6, 15, 31 ) \\ & ( 1, 41, 6, 12, 2 ) & ( 1, 25, 45, 52, 14 ) & ( 1, 14, 2, 19, 48 ) \\ & ( 1, 40, 25, 23, 45 ) & ( 1, 12, 10, 2, 35 ) & ( 1, 22, 14, 44, 2 ) \\ & ( 1, 51, 32, 28, 39 ) & ( 1, 23, 41, 43, 33 ) & ( 1, 10, 22, 27, 41 ) \\ & ( 1, 43, 23, 42, 50 ) & ( 1, 16, 21, 40, 32 ) & ( 1, 48, 24, 4, 27 ) \\ & ( 1, 38, 18, 31, 35 ) & ( 1, 28, 41, 33, 45 ) & ( 1, 30, 44, 49, 39 ) \\ & ( 1, 19, 31, 48, 35 ) & ( 1, 52, 17, 36, 12 ) & ( 1, 6, 1, 30, 31 ) & | \\ & ( 1, 18, 9, 32, 32 ) & ( 1, 37, 32, 3, 31 ) & ( 1, 36, 20, 39, 51 ) \\ & ( 1, 13, 43, 24, 12 ) & ( 1, 1, 1, 1, 51 ) & ( 1, 42, 25, 34, 14 ) \\ & ( 1, 31, 42, 22, 45 ) & ( 1, 3, 32, 13, 50 ) & ( 1, 44, 46, 51, 39 ) \\ & ( 1, 27, 14, 38, 50 ) & ( 1, 49, 3, 7, 19 ) & ( 1, 15, 19, 17, 20 ) \\ & ( 1, 50, 43, 45, 35 ) & ( 1, 46, 14, 25, 32 ) & ( 1, 39, 20, 50, 35 ) \\ & ( 1, 33, 19, 11, 18 ) & ( 1, 35, 31, 46, 26 ) & ( 1, 9, 49, 41, 39 ) \\ & ( 0, 0, 0, 0, 1 ) & ( 1, 34, 1, 10, 30 ) & ( 1, 5, 12, 18, 31 ) \\ & ( 1, 21, 30, 8, 45 ) & ( 1, 32, 31, 29, 19 ) & ( 1, 7, 35, 5, 12 ) \end{aligned} \begin{aligned}X_{12} & ( 1, 49, 15, 31, 5 ) & ( 0, 0, 0, 0, 1 ) & ( 1, 12, 24, 48, 14 ) \\ q = 53 & ( 1, 21, 37, 15, 12 ) & ( 1, 20, 39, 37, 18 ) & ( 1, 52, 42, 46, 31 ) & | \\ & ( 1, 0, 0, 0, 0 ) & ( 1, 41, 26, 30, 2 ) & ( 1, 2, 26, 20, 26 ) \\ & ( 1, 38, 47, 33, 35 ) & ( 1, 45, 23, 39, 48 ) & ( 1, 16, 33, 49, 35 ) \\ & ( 1, 46, 21, 32, 48 ) & ( 1, 43, 14, 7, 33 ) & ( 1, 3, 0, 47, 18 ) \\ & ( 1, 28, 15, 35, 14 ) & ( 1, 8, 14, 1, 8 ) & ( 1, 44, 37, 26, 31 ) & | \\ & ( 1, 9, 30, 51, 19 ) & ( 1, 7, 3, 8, 41 ) & ( 1, 5, 27, 38, 35 ) \\ & ( 1, 51, 12, 43, 48 ) & ( 1, 27, 42, 34, 21 ) & ( 1, 26, 41, 27, 2 ) \\ & ( 1, 48, 15, 29, 26 ) & ( 1, 30, 51, 24, 18 ) & ( 1, 18, 31, 23, 3 ) \\ & ( 1, 47, 22, 13, 18 ) & ( 1, 32, 29, 36, 21 ) & ( 1, 42, 39, 52, 5 ) & | \\ & ( 1, 35, 46, 4, 23 ) & ( 1, 24, 42, 25, 21 ) & ( 1, 36, 37, 17, 33 ) \\ & ( 1, 14, 30, 2, 26 ) & ( 1, 15, 35, 3, 2 ) & ( 1, 22, 11, 6, 12 ) \\ & ( 1, 39, 41, 9, 35 ) & ( 1, 34, 5, 28, 30 ) & ( 1, 10, 33, 10, 30 ) \\ & ( 1, 29, 37, 5, 23 ) & ( 1, 25, 17, 18, 3 ) & ( 1, 1, 4, 19, 18 ) & | \\ & ( 1, 11, 38, 12, 14 ) & ( 1, 17, 15, 16, 33 ) & ( 1, 19, 11, 44, 50 ) \\ & ( 1, 23, 32, 45, 8 ) & ( 1, 40, 51, 42, 12 ) & ( 1, 33, 43, 22, 22 ) \\ & ( 1, 37, 35, 40, 51 ) & ( 1, 31, 3, 11, 21 ) & ( 1, 50, 45, 14, 31 ) \\ & ( 1, 13, 34, 21, 2 ) & ( 1, 6, 40, 50, 8 ) & ( 1, 4, 44, 41, 20 ) \end{aligned}

\begin{aligned}X_{120} & ( 1, 0, 0, 0, 0 ) & ( 1, 20, 23, 24, 53 ) & ( 1, 31, 45, 58, 12 ) \\ q = 59 & ( 1, 5, 41, 43, 51 ) & ( 1, 55, 28, 36, 9 ) & ( 1, 32, 47, 32, 12 ) \\ & ( 1, 34, 19, 51, 29 ) & ( 1, 46, 58, 22, 49 ) & ( 1, 41, 13, 3, 3 ) \\ & ( 1, 18, 54, 50, 19 ) & ( 1, 28, 46, 9, 51 ) & ( 1, 33, 10, 26, 45 ) \\ & ( 1, 54, 19, 47, 51 ) & ( 1, 26, 2, 56, 15 ) & ( 1, 4, 53, 2, 15 ) \\ & ( 1, 49, 20, 35, 9 ) & ( 1, 44, 7, 34, 48 ) & ( 1, 47, 41, 30, 36 ) \\ & ( 1, 17, 50, 29, 16 ) & ( 1, 56, 50, 54, 22 ) & ( 0, 0, 0, 0, 1 ) \\ & ( 1, 11, 55, 49, 35 ) & ( 1, 35, 18, 23, 51 ) & ( 1, 21, 1, 44, 19 ) \\ & ( 1, 23, 37, 6, 27 ) & ( 1, 2, 4, 4, 35 ) & ( 1, 30, 7, 48, 45 ) \\ & ( 1, 51, 5, 33, 3 ) & ( 1, 48, 49, 16, 17 ) & ( 1, 36, 32, 15, 29 ) \\ & ( 1, 14, 19, 18, 36 ) & ( 1, 57, 10, 38, 35 ) & ( 1, 13, 0, 53, 19 ) \\ & ( 1, 58, 53, 55, 19 ) & ( 1, 16, 32, 27, 19 ) & ( 1, 25, 36, 1, 36 ) \\ & ( 1, 27, 1, 20, 49 ) & ( 1, 10, 4, 31, 28 ) & ( 1, 42, 38, 45, 29 ) \\ & ( 1, 19, 38, 25, 28 ) & ( 1, 7, 12, 11, 15 ) & ( 1, 8, 35, 12, 36 ) \\ & ( 1, 50, 5, 19, 28 ) & ( 1, 43, 16, 14, 27 ) & ( 1, 1, 30, 46, 57 ) \\ & ( 1, 6, 31, 8, 53 ) & ( 1, 40, 34, 52, 9 ) & ( 1, 38, 38, 57, 48 ) \\ & ( 1, 29, 12, 17, 12 ) & ( 1, 9, 24, 39, 17 ) & ( 1, 39, 13, 40, 41 ) \\ & ( 1, 24, 18, 37, 27 ) & ( 1, 37, 7, 41, 27 ) & ( 1, 53, 21, 28, 22 ) \\ & ( 1, 52, 40, 42, 51 ) & ( 1, 22, 58, 21, 9 ) & ( 1, 12, 23, 7, 36 ) \\ & ( 1, 3, 58, 13, 19 ) & ( 1, 15, 58, 10, 26 ) & ( 1, 45, 54, 5, 45 ) \end{aligned} \begin{aligned}X_{24} & ( 1, 36, 40, 12, 53 ) & ( 1, 54, 38, 46, 46 ) & ( 1, 5, 25, 4, 45 ) \\ q = 59 & ( 1, 58, 45, 9, 19 ) & ( 1, 41, 13, 21, 26 ) & ( 1, 18, 6, 34, 28 ) \\ & ( 1, 13, 7, 42, 17 ) & ( 1, 55, 7, 58, 28 ) & ( 1, 21, 44, 35, 20 ) \\ & ( 1, 12, 3, 51, 20 ) & ( 1, 8, 48, 38, 5 ) & ( 1, 3, 22, 30, 25 ) & | \\ & ( 1, 25, 38, 50, 41 ) & ( 1, 30, 21, 32, 36 ) & ( 1, 34, 46, 56, 45 ) \\ & ( 1, 19, 1, 31, 45 ) & ( 0, 0, 0, 0, 1 ) & ( 1, 1, 25, 53, 12 ) \\ & ( 1, 24, 28, 2, 51 ) & ( 1, 51, 16, 14, 48 ) & ( 1, 44, 4, 52, 9 ) \\ & ( 1, 11, 1, 5, 48 ) & ( 1, 40, 35, 36, 9 ) & ( 1, 0, 0, 0, 0 ) \\ & ( 1, 33, 54, 11, 29 ) & ( 1, 53, 43, 49, 53 ) & ( 1, 43, 52, 20, 57 ) \\ & ( 1, 28, 35, 18, 1 ) & ( 1, 27, 50, 57, 19 ) & ( 1, 26, 12, 41, 19 ) \\ & ( 1, 45, 17, 26, 41 ) & ( 1, 35, 35, 48, 5 ) & ( 1, 14, 23, 28, 51 ) \\ & ( 1, 39, 31, 43, 15 ) & ( 1, 10, 5, 7, 27 ) & ( 1, 49, 34, 3, 36 ) & | \\ & ( 1, 17, 39, 45, 20 ) & ( 1, 4, 42, 47, 20 ) & ( 1, 2, 36, 40, 9 ) \\ & ( 1, 32, 1, 23, 16 ) & ( 1, 38, 13, 22, 51 ) & ( 1, 7, 16, 44, 41 ) \\ & ( 1, 16, 58, 1, 28 ) & ( 1, 20, 11, 16, 48 ) & ( 1, 56, 45, 39, 9 ) \\ & ( 1, 23, 21, 19, 28 ) & ( 1, 9, 17, 33, 29 ) & ( 1, 42, 16, 17, 48 ) \\ & ( 1, 31, 38, 27, 41 ) & ( 1, 15, 48, 29, 51 ) & ( 1, 46, 18, 13, 29 ) \\ & ( 1, 37, 58, 24, 41 ) & ( 1, 47, 24, 10, 35 ) & ( 1, 48, 37, 25, 36 ) \\ & ( 1, 57, 23, 15, 1 ) & ( 1, 6, 30, 55, 35 ) & ( 1, 22, 1, 54, 36 ) \\ & ( 1, 29, 22, 8, 1 ) & ( 1, 52, 57, 6, 41 ) & ( 1, 50, 18, 37, 16 ) \end{aligned}

X_{5}  ( 1, 0, 0, 0, 0 )  ( 1, 24, 54, 52, 25 )  ( 1, 55, 0, 21, 25 )
q = 59 ( 1, 23, 52, 12, 29 )  ( 1, 3, 36, 35, 15 ) | ( 0, 0, 0, 0, 1 )
( 1, 53, 5, 54, 4 ) ( 1, 19, 28, 27, 1 ) ( 1, 41, 15, 13, 9 )
( 1, 11, 13, 18, 46 ) | ( 1, 10, 30, 10, 3 )  ( 1, 56, 17, 20, 7 )
( 1, 6, 30, 31, 35 )  ( 1, 15, 36, 41, 36 )  ( 1, 43, 21, 44, 27 ) |
( 1, 58, 11, 30, 27 )  ( 1, 8, 35, 16, 4 )  ( 1, 45, 16, 6, 5 )
( 1, 44, 35, 39, 9 )  ( 1, 12, 46, 25, 3 ) | ( 1, 35, 53, 37, 26 )
( 1, 52, 40, 47, 27 ) ( 1, 30, 31, 40, 22 ) ( 1, 51, 15, 58, 3 )
( 1, 34, 15, 34, 35 ) | ( 1, 13, 15, 38, 48 )  ( 1, 21, 4, 49, 17 )
( 1, 7, 14, 45, 20 )  ( 1, 50, 42, 24, 45 )  ( 1, 16, 22, 46, 19 ) |
( 1, 40, 19, 11, 25 )  ( 1, 22, 11, 42, 17 )  ( 1, 29, 15, 2, 12 )
( 1, 5, 33, 29, 5 )  ( 1, 57, 39, 7, 27 ) | ( 1, 14, 19, 5, 41 )
( 1, 49, 39, 15, 45 ) ( 1, 38, 41, 23, 41 ) ( 1, 54, 24, 26, 26 )
( 1, 25, 3, 57, 41 ) | ( 1, 31, 0, 53, 9 )  ( 1, 27, 43, 50, 46 )
( 1, 17, 19, 48, 3 )  ( 1, 26, 42, 55, 51 )  ( 1, 28, 41, 19, 29 ) |
( 1, 1, 30, 32, 12 )  ( 1, 48, 19, 14, 29 )  ( 1, 39, 24, 3, 15 )
( 1, 46, 55, 43, 12 )  ( 1, 18, 50, 28, 5 ) | ( 1, 37, 49, 33, 36 )
( 1, 32, 31, 17, 29 ) ( 1, 42, 1, 8, 17 ) ( 1, 9, 14, 22, 19 )
( 1, 47, 45, 56, 4 ) | ( 1, 4, 12, 1, 29 )  ( 1, 36, 57, 4, 29 )
( 1, 33, 50, 9, 35 )  ( 1, 2, 17, 51, 22 )  ( 1, 20, 20, 36, 1 )

X_{24} ( 1, 40, 64, 64, 18 ) ( 1, 0, 0, 0, 0 )  ( 1, 29, 70, 61, 5 )
q = 71 ( 1, 13, 43, 48, 12 ) ( 1, 44, 22, 47, 4 ) ( 1, 8, 56, 45, 54 )
( 1, 24, 3, 4, 37 )  ( 1, 12, 37, 63, 5 ) ( 1, 48, 2, 62, 2 )
( 1, 68, 13, 59, 8 ) ( 1, 67, 69, 3, 8 )  ( 1, 66, 23, 53, 20 )
( 1, 38, 41, 58, 20 ) ( 1, 46, 14, 11, 8 ) ( 1, 57, 16, 31, 36 )
( 1, 32, 68, 27, 50 ) ( 1, 52, 2, 43, 32 ) ( 1, 37, 37, 33, 37 )
( 1, 39, 26, 42, 29 ) ( 1, 64, 63, 55, 37 ) ( 1, 2, 22, 37, 10 )
( 1, 5, 43, 26, 9 )  ( 1, 25, 70, 39, 18 ) ( 1, 30, 18, 36, 16 ) |
( 1, 20, 7, 20, 48 ) ( 1, 61, 1, 23, 16 ) ( 1, 54, 37, 41, 38 )
( 1, 56, 12, 12, 36 ) ( 1, 1, 35, 51, 2 )  ( 1, 6, 5, 35, 49 )
( 1, 17, 49, 19, 45 ) ( 1, 62, 24, 57, 8 ) ( 1, 23, 26, 29, 6 )
( 1, 59, 9, 65, 60 ) ( 1, 33, 5, 25, 2 )  ( 1, 50, 67, 40, 43 )
( 1, 69, 50, 21, 27 ) ( 1, 49, 44, 30, 2 ) ( 1, 53, 19, 60, 9 )
( 1, 45, 21, 68, 49 ) ( 1, 15, 30, 46, 43 ) ( 1, 41, 48, 44, 10 )
( 1, 19, 27, 52, 58 ) ( 1, 28, 10, 9, 3 )  ( 0, 0, 0, 0, 1 )
( 1, 47, 61, 17, 24 ) ( 1, 31, 7, 28, 6 )  ( 1, 11, 57, 16, 54 ) |
( 1, 7, 5, 6, 4 )  ( 1, 70, 30, 15, 38 ) ( 1, 34, 44, 8, 64 )
( 1, 43, 48, 14, 5 ) ( 1, 22, 70, 49, 57 ) ( 1, 35, 5, 66, 8 )
( 1, 55, 47, 54, 4 ) ( 1, 58, 20, 50, 37 ) ( 1, 51, 5, 56, 30 )
( 1, 60, 2, 69, 45 ) ( 1, 63, 3, 13, 24 ) ( 1, 3, 18, 5, 16 )
( 1, 16, 18, 24, 2 ) ( 1, 4, 36, 7, 25 )  ( 1, 65, 35, 32, 32 )
( 1, 14, 6, 70, 49 ) ( 1, 42, 13, 1, 2 )  ( 1, 10, 51, 38, 1 )
( 1, 26, 28, 67, 30 ) ( 1, 36, 52, 22, 54 ) ( 1, 9, 27, 18, 32 )
( 1, 18, 28, 2, 32 ) ( 1, 27, 20, 10, 40 ) ( 1, 21, 46, 34, 10 )

X_{24} ( 1, 77, 47, 43, 41 ) ( 1, 50, 57, 5, 70 ) ( 1, 82, 59, 2, 7 )
q = 83 ( 1, 6, 67, 54, 1 )  ( 1, 80, 28, 69, 4 ) ( 1, 67, 31, 64, 63 )
( 1, 8, 56, 3, 77 )  ( 1, 18, 14, 77, 78 ) ( 1, 64, 56, 68, 65 )
( 1, 51, 22, 16, 28 ) ( 1, 2, 81, 21, 37 ) ( 1, 39, 31, 19, 41 ) |
( 1, 32, 48, 39, 17 ) ( 1, 0, 0, 0, 0 )  ( 1, 81, 36, 37, 23 )
( 1, 54, 39, 57, 49 ) ( 1, 13, 55, 53, 21 ) ( 1, 70, 71, 6, 17 )
( 1, 47, 26, 58, 1 ) ( 1, 26, 54, 50, 17 ) ( 1, 38, 70, 7, 63 )
( 1, 31, 56, 79, 59 ) ( 1, 49, 63, 74, 41 ) ( 1, 4, 48, 20, 23 )
( 1, 44, 33, 1, 29 ) ( 1, 42, 26, 63, 81 ) ( 1, 60, 72, 17, 21 )
( 1, 24, 4, 22, 37 ) ( 1, 73, 49, 80, 59 ) ( 1, 40, 32, 33, 63 )
( 1, 57, 28, 47, 23 ) ( 1, 55, 76, 76, 4 ) ( 1, 9, 27, 26, 33 )
( 1, 3, 20, 4, 3 )  ( 1, 19, 13, 78, 9 ) ( 1, 14, 4, 31, 48 ) |
( 1, 69, 16, 9, 36 ) ( 1, 41, 33, 11, 37 ) ( 1, 33, 56, 60, 30 )
( 1, 76, 0, 81, 69 ) ( 1, 56, 17, 32, 77 ) ( 1, 68, 35, 25, 63 )
( 1, 37, 5, 67, 69 ) ( 1, 71, 50, 35, 78 ) ( 1, 58, 38, 29, 28 )
( 1, 78, 51, 62, 33 ) ( 1, 17, 54, 66, 29 ) ( 1, 35, 48, 59, 30 )
( 1, 43, 39, 23, 63 ) ( 1, 7, 37, 8, 69 )  ( 1, 21, 7, 41, 3 )
( 1, 72, 42, 27, 27 ) ( 1, 79, 65, 75, 59 ) ( 1, 15, 12, 10, 38 )
( 1, 48, 13, 30, 51 ) ( 0, 0, 0, 0, 1 )  ( 1, 22, 2, 15, 81 )
( 1, 30, 62, 73, 25 ) ( 1, 53, 32, 61, 59 ) ( 1, 20, 60, 45, 65 ) |
( 1, 1, 47, 40, 75 ) ( 1, 12, 61, 34, 21 ) ( 1, 63, 54, 52, 33 )
( 1, 29, 57, 46, 65 ) ( 1, 11, 54, 65, 21 ) ( 1, 25, 17, 12, 75 )
( 1, 62, 40, 36, 69 ) ( 1, 46, 65, 51, 69 ) ( 1, 27, 39, 44, 30 )
( 1, 5, 63, 28, 41 ) ( 1, 16, 28, 56, 63 ) ( 1, 45, 0, 38, 33 )
( 1, 66, 25, 48, 25 ) ( 1, 61, 71, 13, 59 ) ( 1, 36, 49, 72, 70 )
( 1, 28, 41, 82, 7 ) ( 1, 75, 60, 55, 77 ) ( 1, 34, 55, 42, 29 )
( 1, 10, 63, 49, 23 ) ( 1, 65, 1, 24, 16 ) ( 1, 52, 23, 71, 12 )
( 1, 23, 20, 14, 25 ) ( 1, 74, 17, 70, 9 ) ( 1, 59, 14, 18, 70 )

X_{24} ( 1, 0, 0, 0, 0 )  ( 1, 22, 8, 4, 26 )  ( 1, 54, 17, 25, 52 )
q = 89 ( 1, 79, 36, 40, 83 ) ( 1, 30, 69, 30, 35 ) ( 1, 43, 85, 81, 61 ) |
( 1, 25, 55, 29, 77 ) ( 1, 56, 28, 2, 83 ) ( 1, 6, 34, 36, 52 )
( 1, 29, 2, 15, 6 ) ( 1, 3, 50, 84, 7 ) ( 1, 61, 64, 87, 31 )
( 1, 10, 20, 14, 83 ) ( 1, 13, 0, 45, 38 ) ( 1, 11, 88, 31, 14 )
( 1, 20, 47, 11, 63 ) ( 1, 74, 26, 33, 86 ) ( 1, 49, 37, 3, 86 ) |
( 1, 16, 57, 55, 54 ) ( 0, 0, 0, 0, 1 )  ( 1, 40, 83, 22, 63 )
( 1, 23, 38, 71, 38 ) ( 1, 12, 63, 70, 86 ) ( 1, 21, 7, 19, 86 )
( 1, 14, 0, 18, 15 ) ( 1, 45, 42, 39, 41 ) ( 1, 63, 42, 8, 46 )
( 1, 59, 74, 72, 66 ) ( 1, 8, 38, 61, 26 ) ( 1, 51, 19, 7, 83 )
( 1, 19, 68, 83, 29 ) ( 1, 70, 47, 86, 48 ) ( 1, 4, 51, 53, 35 )
( 1, 67, 0, 41, 12 ) ( 1, 62, 25, 65, 62 ) ( 1, 68, 43, 28, 74 )
( 1, 2, 67, 77, 74 ) ( 1, 26, 83, 17, 56 ) ( 1, 35, 66, 49, 70 )
( 1, 83, 20, 62, 61 ) ( 1, 84, 18, 20, 43 ) ( 1, 66, 56, 38, 52 ) |
( 1, 86, 27, 48, 38 ) ( 1, 60, 34, 68, 15 ) ( 1, 44, 12, 6, 37 )
( 1, 75, 78, 23, 23 ) ( 1, 73, 11, 50, 56 ) ( 1, 80, 70, 79, 83 )
( 1, 28, 57, 76, 52 ) ( 1, 27, 23, 78, 35 ) ( 1, 53, 17, 88, 31 )
( 1, 39, 64, 54, 28 ) ( 1, 88, 9, 51, 59 ) ( 1, 78, 55, 64, 82 )
( 1, 15, 73, 32, 65 ) ( 1, 87, 52, 60, 86 ) ( 1, 72, 9, 13, 51 )
( 1, 81, 60, 42, 29 ) ( 1, 48, 60, 10, 14 ) ( 1, 52, 67, 58, 60 )
( 1, 38, 82, 34, 83 ) ( 1, 64, 34, 56, 66 ) ( 1, 31, 55, 27, 54 )
( 1, 55, 77, 37, 46 ) ( 1, 9, 26, 69, 38 ) ( 1, 47, 9, 67, 63 ) |
( 1, 18, 64, 16, 66 ) ( 1, 24, 17, 46, 31 ) ( 1, 71, 50, 26, 15 )
( 1, 32, 49, 66, 26 ) ( 1, 37, 61, 43, 28 ) ( 1, 58, 25, 35, 15 )
( 1, 41, 21, 12, 46 ) ( 1, 42, 25, 59, 12 ) ( 1, 34, 25, 74, 63 )
( 1, 69, 33, 44, 58 ) ( 1, 57, 35, 73, 43 ) ( 1, 36, 46, 24, 46 )
( 1, 77, 14, 85, 23 ) ( 1, 65, 25, 57, 31 ) ( 1, 85, 67, 9, 86 )
( 1, 50, 52, 80, 60 ) ( 1, 7, 55, 52, 82 ) ( 1, 5, 59, 75, 60 )
( 1, 17, 24, 21, 46 ) ( 1, 82, 74, 47, 15 ) ( 1, 46, 72, 5, 15 )
( 1, 76, 7, 82, 38 ) ( 1, 33, 33, 1, 35 ) ( 1, 1, 71, 63, 58 )

8.2 The new example for q = 125

The techniques discussed in the previous chapter rely heavily upon the implementation in MAGMA of the groups P GO(5, q). As q gets large, say q > 59, the techniques are no longer as effective in terms of the computing time required, and for q > 97 we no longer have available these in-built groups. Thus to search over these higher order fields, we may start with certain hypotheses upon the examples we wish to find, or instead change techniques altogether. For example, the three examples found for q = 71, 83 and 89 required a hypothesis concerning the group of the example, as we indicate in the final chapter. Motivated by the construction of an infinite family of BLT-sets from an example for q = 27, which we discuss in the final chapter also, an attempt was made to construct BLT-sets for q = 125. Working with the underlying vector spaces, matrix groups and the classical BLT-set over GF (5), the following example was discovered.

Let q = 125 and define Q(4, q) by the quadratic form Q : x_{0}x_{5} + x_{2}x_{4} + x_{3}^{2}. Then the following defines a BLT-set of Q(4, 125).

X_{72} = {⟨(1, t, f(t), −g(t), tg(t) − f(t)^{2})⟩ : t in GF (q)} ∪ {⟨(0, 0, 0, 0, 1)⟩}

where f and g are as follows :

f = x^{59} + 2x^{57} + 4x^{55} + 2x^{53} + x^{51} + 3x^{49} + 4x^{45} + 4x^{43} + 4x^{39} + 2x^{37}
+4x^{35} + 3x^{33} + 4x^{31} + 4x^{29} + 3x^{25} + 4x^{23} + x^{19} + 4x^{17} + 4x^{15} + 4x^{13}
+3x^{11} + 2x^{9} + x^{7} + 4x^{5} + 4x
g = x^{117} + 3x^{115} + x^{113} + x^{107} + 3x^{105} + 3x^{103} + 3x^{101} + 3x^{95} + 2x^{91} + 4x^{87}
+2x^{85} + 3x^{83} + 2x^{77} + 2x^{75} + 4x^{73} + 4x^{71} + x^{67} + 2x^{65} + 2x^{61} + 4x^{59}
+x^{57} + x^{55} + 4x^{53} + 4x^{49} + 3x^{47} + 2x^{45} + 4x^{43} + 2x^{41} + 4x^{39} + x^{33}
+2x^{31} + 4x^{29} + 3x^{25} + 4x^{23} + x^{21} + x^{19} + 4x^{17} + x^{15} + 3x^{13} + 4x^{11}
+2x^{7} + 4x^{3} + 4x.

To determine the group of this example, an adaptation of the conic invariant of Williams (1998) was used. This involves counting the number of points of a BLTset which lie on the conic in the plane spanned by three distinct elements of the BLT-set, for all triples in the BLT-set. From this, we have a unique conic C meeting X_{72} in 6 points, these being the points over GF (5), and all others meeting X_{72} in at most 4 points. Hence the stabiliser H of X_{72} in P Γ O(5, 125) is contained in the stabiliser G of X_{72} ∩ C in P Γ O(5, 125). Since |G| = 4320, we can loop over

8.2. The new example for q = 125  119

all elements of G and see if they stabilise X_{72} in order to determine H. We find that |H| = 72, which distinguishes X_{72} from the known examples over q = 125. Furthermore, H ≅ S_{4} × C_{3}, with orbit structure {72, 24^{2}, 6}. The four orbits are actually distinguished by how many conics meeting X_{72} in 4 points a point lies on. Noting that the points of the BLT-set other than (∞) = ⟨(0, 0, 0, 0, 1)⟩ are indexed by the elements of GF (125) (with respect to the second coordinate), we describe the orbits on X_{72} as a partition of GF (125) ∪ {∞}, where w^{3} = 2w + 2 :

{0, 1, 2, 3, 4, (∞)},

\begin{matrix}\{w^{6}, w^{19}, w^{21}, w^{26}, w^{29}, w^{30}, w^{33}, w^{34}, w^{41}, w^{43}, w^{44}, w^{46}, w^{68}, w^{81}, w^{83}, w^{88}, \\ w^{91}, w^{92}, w^{95}, w^{96}, w^{103}, w^{105}, w^{106}, w^{108}\}, \end{matrix}

\begin{matrix}\{w^{7}, w^{9}, w^{17}, w^{23}, w^{35}, w^{37}, w^{39}, w^{45}, w^{51}, w^{53}, w^{57}, w^{61}, w^{69}, w^{71}, w^{79}, w^{85}, \\ w^{97}, w^{99}, w^{101}, w^{107}, w^{113}, w^{115}, w^{119}, w^{123}\}, \end{matrix}

{w, w^{2}, w^{3}, w^{4}, w^{5}, w^{8}, w^{10}, w^{11}, w^{12}, w^{13}, w^{14}, w^{15}, w^{16}, w^{18}, w^{20}, w^{22}, w^{24}, w^{25}, w^{27}, w^{28}, w^{32}, w^{36}, w^{38}, w^{40}, w^{42}, w^{47}, w^{48}, w^{49}, w^{50}, w^{52}, w^{54}, w^{55}, w^{56}, w^{58}, w^{59}, w^{60}, w^{63}, w^{64}, w^{65}, w^{66}, w^{67}, w^{70}, w^{72}, w^{73}, w^{74}, w^{75}, w^{76}, w^{77}, w^{78}, w^{80}, w^{82}, w^{84}, w^{86}, w^{87}, w^{89}, w^{90}, w^{94}, w^{98}, w^{100}, w^{102}, w^{104}, w^{109}, w^{110}, w^{111}, w^{112}, w^{114}, w^{116}, w^{117}, w^{118}, w^{120}, w^{121}, w^{122}}.

120  Chapter 8. The new BLT-sets

CHAPTER 9 Analysis of some examples

One of the important motivations behind searching for new examples of BLT-sets is the possibility of generalising examples to infinite families of BLT-sets. Of course, some examples may not belong to an infinite family of BLT-sets, and so we also look for evidence which suggests that an example is sporadic (as a BLT-set). For example, the three new examples for q = 71, 83 and 89 presented in the previous chapter were discovered using similar orbit-stitching techniques as discussed earlier, but required starting with a particular subgroup. Analysis of some other examples suggested the possibility of further examples arising from a subgroup of the stabiliser of the Fisher-Thas-Walker/Kantor BLT-set isomorphic to A_{4}. The recent discovery of these examples, working with this hypothesis, highlights the opportunities available from analysing known examples, and why it is worthwhile searching for new examples in the first place. In this final chapter we consider some of the other examples presented earlier. In particular, we have been able to generalise the new example for q = 27 to an infinite family of BLT-sets in characteristic 3.

9.1 An infinite family of BLT-sets in characteristic 3

We have seen in Section 6.3 that BLT-sets may be parametrised in the same manner as the associated q-clans and flocks. Thus given coordinates for the points of a BLT-set, as for example in the previous chapter, we may attempt to find such a parametrisation which is useful in describing the example in more general terms.

We do this by first moving the BLT-set onto the points P_{∞} = (0, 0, 0, 0, 1) and P_{0} = (1, 0, 0, 0, 0), and then, for the points of the BLT-set other than P_{∞}, interpolating the second and third coordinates and the second and fourth coordinates to obtain polynomial functions describing the BLT-set. These can then be translated into the q-clan model, and the condition on the matrices of the q-clan can be easily checked. However, this process is far from unique. There is a choice of which points to move to the special points, and we may also then change coordinates in any way that fixes these points, before interpolating, and consequently we cannot afford to check all possible interpolations. Instead, we hope to find an interpolation which provides a form which can be generalised in some manner, depending upon the situation.

Such experimentation was successfully performed on the new example X_{6} for q = 27. This example has two fixed points, an orbit of length 2 and four of length 6 under its stabiliser P Γ O(5, q)_{X_{6}}. As a general rule of thumb, we choose a point in the shortest orbit to map to the point P_{∞}, and then choose a point in the shortest orbit of P Γ O(5, q)_{X_{6},P_{∞}} on X_{6}\{P_{∞}} to map to P_{0} in order to preserve as much symmetry of X_{6} as possible. We also choose to change coordinates in such a way that the semisimilarity

(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) ↦ (x_{0}^{3}, x_{1}^{3}, x_{2}^{3}, x_{3}^{3}, x_{4}^{3})

preserves X_{6} (which now contains P_{∞} and P_{0}) since then the functions appearing in the q-clan will be polynomial over GF (3). Interpolating as above, and a bit of luck, led to a q-clan for which −det(A_{s} − A_{t}) = −S(s, t), where S(s, t) was the square of a polynomial over GF (3). As −1 is a nonsquare in GF (3), −det(A_{s} − A_{t}) remains equal to −S(s, t) for all odd degree extensions of GF (3), and so gives an infinite family of q-clans for q = 3^{h}, h odd. Motivated by the known infinite families which are described for a particular characteristic only (namely, Kantor/Knuth semifield, Kantor likeable, and Ganley semifield), we can generalise to the use of an arbitrary nonsquare in place of −1 and then attempt to remove the hypothesis of h odd. This led to the following infinite family of BLT-sets in characteristic 3, described using the q-clan model.

Theorem 9.1. For t ∈ GF (q) with q = 3^{h}, and n a fixed nonsquare of GF (q), let

A_{t} = [ \begin{matrix}t & t^{4} + nt^{2} \\ t^{4} + nt^{2} & −n^{−1}t^{9} + t^{7} + n^{2}t^{3} − n^{3}t \end{matrix} ] .

Then C = {A_{t} : t ∈ GF (q)} is a q-clan.

Proof :

(b_{s} − b_{t})^{2} − 4(a_{s} − a_{t})(c_{s} − c_{t})
= (−s^{4} − ns^{2} + t^{4} + nt^{2})^{2}
−4(s − t)(−n^{−1}s^{9} + s^{7} + n^{2}s^{3} − n^{3}s + n^{−1}t^{9} − t^{7} − n^{2}t^{3} + n^{3}t)
= (s − t)^{2}(s + t)^{2}(n + s^{2} + t^{2})^{2}
+n^{−1}(s − t)^{2}(s^{8} + s^{7}t + s^{6}t^{2} + s^{5}t^{3} + s^{4}t^{4} + s^{3}t^{5} + s^{2}t^{6} + st^{7} + t^{8}
−n(s^{6} + s^{5}t + s^{4}t^{2} + s^{3}t^{3} + s^{2}t^{4} + st^{5} + t^{6}) − n^{3}(s^{2} + st + t^{2}) + n^{4})
= n^{−1}(s − t)^{2}(s^{8} + s^{7}t + s^{6}t^{2} + s^{5}t^{3} + s^{4}t^{4} + s^{3}t^{5} + s^{2}t^{6} + st^{7} + t^{8}

+n(s^{5}t + 2s^{4}t^{2} + 2s^{2}t^{4} + st^{5}) + n^{2}(2s^{4} + s^{3}t + s^{2}t^{2} + st^{3} + 2t^{4}) + n^{3}st + n^{4})

= n^{−1}(s − t)^{2}(s^{4} − s^{3}t − st^{3} + t^{4} − nst + n^{2})^{2}

We must also show that this is nonzero for s ≠ t. This is if and only if (s − t)^{4} − nst + n^{2} is nonzero. So suppose (s − t)^{4} − nst + n^{2} = 0. Then

n = −st ± \sqrt{s^{2}t^{2} − (s − t)^{4}}
= −st ± (s + t)\sqrt{−(s^{2} + t^{2})}
= ((s + t) ∓ \sqrt{−(s^{2} + t^{2})})^{2}.

contradicting the fact that n is a nonsquare.

For q = 3 the BLT-set in the family is classical, and for q = 9 it is Fisher. We know from the work in the previous chapter that the example for q = 27 is new, and suspected that the family was new in general for q > 27. The infinite family was published in a paper by Penttila and the present author (2001), with a conjecture that the full stabiliser of the BLT-set for q = 3^{h} > 9 is cyclic of order 2h. By the so-called Orbit Counting Lemma, this would imply that at least \frac{q+1}{2h} distinct, non-isomorphic flocks arise from each BLT-set, unlike all previously known families which had at most two flocks arising in any case.

This was indeed proven to be the case by Payne (2001) who calculated the collineation group of the associated GQ using the Fundamental Theorem of q-clan geometry (see Chapter 5). Hence the number of distinct flocks arising from each BLT-set grows with the field order q. Furthermore, if h is the power of an odd prime, Payne (2001) shows that many of the flocks arising have trivial automorphism group. This provides the first infinite family of such flocks.

9.2 Possibly sporadic examples

As discussed above, some examples of BLT-sets admit evidence that they may not belong to an infinite family of BLT-sets, and so we say that these BLT-sets are possibly sporadic (as BLT-sets). The example of Penttila and Royle P R_{1152} over the field of order 23, and the example of De Clerck and Penttila DCP_{2304} over the field of order 47, have arisen in a study of hyperbolic reflections preserving a BLT-set of Q(4, q), by Bader, Durante, Lunardon, Penttila and the present author ([8]). In this paper, these two BLT-sets are linked with two of the exceptional flocks of the hyperbolic quadric (Section 3) via root systems of the Euclidean space ℝ^{4}, providing evidence that they are indeed sporadic as BLT-sets.

A further example in the data of the previous chapter appears to be possibly sporadic as a BLT-set. We give here a description of the new example X_{720} over the field of order 29, which suggests that this may be the case.

Let V be GF (29)^{6} and Q be the quadratic form on V given by

Q(x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2}.

Then Q(1, 1, 1, 1, 1, 1) is not 0, so W = {x|x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} = 0} is a nondegenerate subspace. Let P be the set of all images of the point (1, 1, 1, 1, 7, 18) under permutations of the coordinates. Then P is a set of 30 points of W. We now prove that P is a BLT-set of W (considered as O(5, 29)) whose group is S_{6}. Theorem 9.2. Let V = GF (29)^{6} and Q: V → GF (29) be given by

Q(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2}.

Let e = (1, 1, 1, 1, 1, 1) and W = e^{⊥}, so that W is a nondegenerate subspace of V. Let S_{6} act on V by permuting the coordinates. Let P be the orbit of (1, 1, 1, 1, 7, 18) under S_{6}. Then P is a BLT-set of W, and the full stabiliser of P in P GO(5, 29) is S_{6}.

Proof : The polar form of Q is f(x, y) = 2x_{1}y_{1}+2x_{2}y_{2}+2x_{3}y_{3}+2x_{4}y_{4}+2x_{5}y_{5}+2x_{6}y_{6}. S_{6} fixes W and f(e, (1, 1, 1, 1, 7, 18)) = 0(mod 29), so P is a subset of W. A calculation shows that the discriminant of the restriction of Q to W is a nonsquare, so by Corollary 6.2 we must show that f(x, y)f(x, z)f(y, z) is a nonsquare, for all distinct x, y, z ∈ P. It is sufficient to show that f(x, y) is a nonsquare, for distinct x, y ∈ P. There are 5 cases :

f(x, y) = 2(18^{2} + 7.1 + 1.7 + 1^{2} + 1^{2} + 1^{2}) = 15(mod 29); f(x, y) = 2(18.7 + 7.18 + 1^{2} + 1^{2} + 1^{2} + 1^{2}) = 19(mod 29); f(x, y) = 2(18.7 + 7.1 + 1.18 + 1^{2} + 1^{2} + 1^{2}) = 18(mod 29); f(x, y) = 2(18.1 + 7.7 + 1.18 + 1^{2} + 1^{2} + 1^{2}) = 2(mod 29); f(x, y) = 2(18.1 + 7.1 + 1.18 + 1.7 + 1^{2} + 1^{2}) = 17(mod 29).

These are all nonsquares modulo 29, so P is a BLT-set. P admits S_{6}, which is a maximal subgroup of P GO(5, 29), and so the full stabiliser of P in P GO(5, 29) is S_{6}.

The previously known BLT-sets in Q(4, 29) are the classical BLT-set, with a group of order 1461600, the Fisher-Thas/Walker/Kantor BLT-set, with a group of order 24360, the Fisher BLT-set, with a group of order 1800, and the Penttila BLT-set, with a group of order 60 (which can be calculated using MAGMA). Hence P is a new BLT-set.

Since S_{6} acts transitively on P, there arises exactly one flock of the quadratic cone of PG(3, 29).

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Index

4-gonal family, 38 k-arc, 19 k-cap, 21 q-clan, 52

classical, 52 equivalent, 56 Kantor/Payne monomial, 53

affine plane, 27

order, 27 anisotropic matrix, 51 anisotropic subspace, 7, 9

base point, 35 bilinear form, 4 BLT-set, 70, 72, 75

classical, 70, 80 Fisher, 88 Fisher-Thas/Walker/Kantor, 81 Ganley semifield, 84 Kantor likeable, 84 Kantor monomial, 83 Kantor/Knuth semifield, 82 partial, 71, 72 Penttila, 88

collineation, 2, 35 axial,axis, 28 central,centre, 28

collineation group, 2 companion automorphism, 2, 4 congruence partition, 29 conic, 19 constant of similitude, 6 correlation, 3 correlation group, 3 discriminant, 17 dual grid, 32 duality, 3

elation, 28, 35

elation generalised quadrangle, 35 elation group, 35 equivalent matrices, 52

flag, 25 flock, 39 flock of hyperbolic quadric, 40

exceptional, 40

linear, 40 Thas, 40 flock of ovoid, 39 linear, 40

flock of quadratic cone, 41 derived, 70 Fisher-Thas/Walker, 43 isomorphic, 44 likeable, 71 linear, 41 partial, 71 semifield, 51, 71

generalised n-gon, 25

generalised quadrangle, 15, 32 dual, 32 order, 32

grid, 32

homogeneous coordinates, 27 homology, 28 hyperbolic fibration, 58 back half, 58 front half, 58 \underline{136 Index}

regular, 58 hyperbolic line, 10 hyperbolic pair, 10 hyperbolic space, 12 hyperoval, 19 hyperplane, 1

incidence structure, 25

isomorphic, 26 isometric, 6 isometry, 6 isometry group, 6 isotropic, 7

Knarr construction, 75

likeable function, 46 linespread, 45

isomorphic, 46 regular, 45 regular conical cover, 56 regular elliptic cover, 58

middle nucleus, 30

opposite regulus, 20 orthogonal, 5 orthogonal complement, 5 orthogonal direct sum, 10 orthogonal space, 7

elliptic, 13 hyperbolic, 13

oval, 19 oval cone, 21 ovoid, 21, 49

Kantor/Knuth semifield, 50

polar form, 8 polar rank, 14 polar space, 3, 7–9 abstract, 14 all-or-one axiom, 14 classical, 9 Tits’ axiom, 14

polar subspace, 4

polarity, 3 null, 7 orthogonal, 7 pseudo-symplectic, 7 symplectic, 7 unitary, 7 Principle of Duality, 3 projective completion, 27 projective geometry, 1

automorphism group, 2 order, 2 projective plane, 26 order, 26

projective semilinear group, 3 projective space, 1 Desarguesian, 2

quadratic cone, 21 quadratic form, 8

nondegenerate, 8 quadric, 8

elliptic, 19 hyperbolic, 19 Klein, 22 nondegenerate, 8

quasifield, 30

radical, 4 regulus, 20

semifield, 30 Ganley, 31 \underline{Index 137}

semilinear transformation, 2 semisimilarity, 6 semisimilarity group, 6 sesquilinear form, 4

alternating, 7

duality arising from, 5 hermitian, 7 matrix of, 4 nondegenerate, 4 reflexive, 5 symmetric, 7

similar, 6 similarity, 6 similarity group, 6 singular, 8 singular radical, 8 skew lines, 19 space with form, 9 spread, 29, 45 spread set, 48 subspace, 1, 14

nondegenerate, 5 symplectic space, 7

ternary field, 27 totally isotropic, 3, 7 totally singular, 8 translation, 28 translation affine plane, 28

kernel, 29 translation group, 28 translation line, 28 translation ovoid, 51 translation plane, 28

Kantor likeable, 47 likeable, 47 semifield, 30 transversal, 19

unitary space, 7

vertex, 21

whorl, 35 Witt index, 10