On some designs and codes invariant under the Higman-Sims group

Abstract

We examine a design V and a binary code C constructed from the first rank-5 primitive permutation representation of degree 1100 of the sporadic simple group HS of Higman-Sims. We prove in Section 5 that Aut(C) = Aut(D) = HS:2 and determine the weight distribution of the code and that of its dual. In Subsection 6.1, we show that for a word w_l of weight l, where l ⋯; {480, 512, 672} the stabilizer (HS)w_l is a maximal subgroup of HS. The words of weight 512 split into two orbits namely W₍₅₁₂₎₁ and W₍₅₁₂₎₂ respectively, and for w_l⋯ {W ₍₅₁₂₎₁, W₍₅₁₂₎₂}, we prove that (HS)_wl is a maximal subgroup of HS. Furthermore in Subsection 6.2 we determine the structures of the stabilizers (HS:2)_wl by extending the results of Subsection 6.1 to HS:2.