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

Document Type: Research Paper

Author

School of Mathematics‎, ‎Statistics and Computer Science‎, ‎University of KwaZulu-Natal‎, ‎Durban 4000‎, ‎South Africa.

Abstract

‎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.

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Main Subjects

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### History

• Receive Date: 31 May 2016
• Revise Date: 20 December 2016
• Accept Date: 15 January 2017