2-modular representations of the alternating group A_8 as binary codes

Through a modular representation theoretical approach we enumerate all non-trivial codes from the 2-modular representations of A8, using a chain of maximal submodules of a permutation module induced by the action of A8 on objects such as points, Steiner S(3,4,8) systems, duads, bisections and triads. Using the geometry of these objects we attempt to gain some insight into the nature of possible codewords, particularly those of minimum weight. Several sets of non-trivial codewords in the codes examined constitute single orbits of the automorphism groups that are stabilized by maximal subgroups. Many self-orthogonal codes invariant under A8 are obtained, and moreover, 22 optimal codes all invariant under A8 are constructed. Finally, we establish that there are no self-dual codes of lengths 28 and 56 invariant under A8 and S8 respectively, and in particular no self-dual doubly-even code of length 56

Codes from adjacency matrices of uniform subset graphs

Studies of the p-ary codes from the adjacency matrices of uniform subset graphs Γ(n,k,r)Γ(n,k,r) and their reflexive associates have shown that a particular family of codes defined on the subsets are intimately related to the codes from these graphs. We describe these codes here and examine their relation to some particular classes of uniform subset graphs. In particular we include a complete analysis of the p-ary codes from Γ(n,3,r)Γ(n,3,r) for p≥5p≥5 , thus extending earlier results for p=2,3p=2,3

Binary codes from graphs on triples

www.elsevier.com/locate/disc For a set of size n ¿ 7 and {3} the set of subsets of of size 3, we examine the binary codes obtained from the adjacency matrix of each of the three graphs with vertex set {3}, with adjacency de ned by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively