Linear codes with complementary duals related to the complement of the Higman-Sims graph

‎In this paper we study codes Cp(HiS^-) where p=3,7‎,‎11 defined by the 3‎- ‎7‎- ‎and 11-modular representations of the simple sporadic group HS of Higman and Sims of degree 100‎. ‎With exception of p=11 the codes are those defined by the row span of the adjacency matrix of the complement of the Higman-Sims graph over GF(3) and GF(7). We show that these codes have a similar decoding performance to that of their binary counterparts obtained from the Higman-Sims graph‎. ‎In particular‎, ‎we show that these are linear codes with complementary duals‎, ‎and thus meet the asymptotic Gilbert-Varshamov bound‎. ‎Furthermore‎, ‎using the codewords of weight 30 in Cp(HiS^-) we determine a subcode of codimension 1‎, ‎and thus show that the permutation module of dimension 100 over the fields of 3‎, ‎7 and 11-elements‎, ‎respectively is the direct sum of three absolutely irreducible modules of dimensions 1‎, ‎22 and 77‎. ‎The latter being also the subdegrees of the orbit decomposition of the rank-3 representation‎.