%0 Journal Article %T Linear codes with complementary duals related to the complement of the Higman-Sims graph %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Rodrigues, B.G. %D 2017 %\ 12/30/2017 %V 43 %N 7 %P 2183-2204 %! Linear codes with complementary duals related to the complement of the Higman-Sims graph %K Strongly regular graph‎ %K ‎Higman-Sims graph‎ %K ‎linear code‎ %K ‎automorphism group‎ %R %X ‎In this paper we study codes $C_p(\overline{{\rm HiS}})$ where $p =3,7‎, ‎11$ defined by the 3‎- ‎7‎- ‎and 11-modular representations of the simple sporadic group ${\rm HS}$ 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 $C_p(\overline{{\rm HiS}})$ 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. %U http://bims.iranjournals.ir/article_1253_82f0207ea7f159c1fbcdbf2bda09cf56.pdf