Rodrigues, B. (2017). Linear codes with complementary duals related to the complement of the Higman-Sims graph. Bulletin of the Iranian Mathematical Society, 43(7), 2183-2204.

B.G. Rodrigues. "Linear codes with complementary duals related to the complement of the Higman-Sims graph". Bulletin of the Iranian Mathematical Society, 43, 7, 2017, 2183-2204.

Rodrigues, B. (2017). 'Linear codes with complementary duals related to the complement of the Higman-Sims graph', Bulletin of the Iranian Mathematical Society, 43(7), pp. 2183-2204.

Rodrigues, B. Linear codes with complementary duals related to the complement of the Higman-Sims graph. Bulletin of the Iranian Mathematical Society, 2017; 43(7): 2183-2204.

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

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

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

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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