On Some Combinatorial Structures Constructed from the Groups , , and

Abstract

We describe a construction of primitive 2-designs and strongly regular graphs from the simple groups , and . The designs and the graphs are constructed by defining incidence structures on conjugacy classes of maximal subgroups of , and . In addition, from the group , we construct 2-designs with parameters and having the full automorphism group isomorphic to .

Some irreducible 2-modular codes invariant under the symplectic group S_6(2)

“…(i) The designs 2-(28, 12, 11) given in Section 6 and D w16 are respectively the derived and the residual designs of a 2-(64, 28, 12) design and they are part of a infinite family of quasi-symmetric designs constructed from the symplectic group S 2m (2) and quadratic forms, see [13,21] Table 4 we deduce that the non-zero weight codewords of the codes C 36,i 1 ≤ i ≤ 3 are single orbits and are stabilized by maximal subgroups of the automorphism groups. We consider the action of Aut(C) = S 6 (2) on the codewords of minimum weight to describe the structure of the stabilizers and form 2-designs which are invariant under S 6 (2).…”

“…(iii) The designs with parameters 2-(28, 10, 40), 2-(36, 12, 33), and 2-(36, 6,8) where first obtained in [13]. The authors queried in that paper whether or not such designs were known to exist.…”

“…From the fixed points of a Sylow 3-subgroup of 2 4 :S 5 one can construct the 2-(28, 10, 40) design. Furthermore, the 2-(36, 6, 8) design is defined by the ovals of the 2- (36, 8, 6) design, see [13]. Here, we used the codes and the supports of codewords of given non-zero weight to show yet another way of constructing such designs and also provide geometric interconnections which uncover the interplay between coding theory and design theory via modular representation theory.…”

“…The said method emerges as a combination of techniques described in [5,6] and partially in [20]. In this article, motivated by questions raised in [11] concerning with a certain class of 2-(64, 28, 12) designs whose derived designs are not quasi-symmetric, and by the work of Crnković and Mikulić ( [13]) we examine binary codes obtained from the permutation modules induced by the action of the simple projective symplectic group S 6 (2) on the cosets of some of its maximal subgroups. Using a chain of maximal submodules of a permutation module induced by the action of the group S 6 (2) on various geometrical objects described in [12] as O − 6 (2), O + 6 (2), points, G 2 (2), isotropic planes, isotropic lines, non-isotropic lines and S 2 (8), we determine all the 2-modular binary linear codes (up to length 135) invariant under the action of the symplectic group S 6 (2).…”

Untitled

“…(i) The designs 2-(28, 12, 11) given in Section 6 and D w16 are respectively the derived and the residual designs of a 2-(64, 28, 12) design and they are part of a infinite family of quasi-symmetric designs constructed from the symplectic group S 2m (2) and quadratic forms, see [13,21]. These designs are on v = 2 2m−1 ± 2 m−1 points depending on whether we consider hyperbolic or elliptic quadratic forms.…”