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);;