Recently, Frank De Clerck, Nicola Durante and I have discovered an interesting hemisystem of the Fisher-Thas-Walker-Kantor generalised quadrangle admitting AGL(1,5) x S3, and it gives rise to....
An interesting hemisystem
an antipodal cometric 4-class association scheme
- the four nontrivial relations are: (i) both in the same half and collinear, (ii) both in the same half and not collinear, (iii) in different halves and collinear, (iv) in different halves and not collinear.
- then in GAP do...
a partial quadrangle PQ(2, 25, 8)
- the points of the PQ are just the elements of the hemisystem; the lines of the PQ are the lines of the ambient GQ.
- then do...
a strongly regular graph srg(378,52,1,8)
- the point-graph of the above PQ
- then do...
Read(“AS.txt”);
a0 := IdentityMat(Rationals, 756);;
allones := 0*a0+1;;
a0+a1+a2+a3+a4 = allones;
pairs := Combinations([a1,a2,a3,a4],2);;
ForAll(pairs, t -> t[1]*t[2] = t[2]*t[1]);
vec := VectorSpace(Rationals,[a0,a1,a2,a3,a4]);
ForAll(pairs, t -> t[1]*t[2] in vec);
Read(“PQ.txt”);
LoadPackage(“Design”);
design := BlockDesign(378, blocks);;
Read(“SRG.txt”);
LoadPackage(“Grape”);
adj := function(i, j)
return i<>j and mat[i][j] := 1;
end;
graph := Graph(Group(()),[1..378],OnPoints,adj);;