Self-orthogonal codes constructed from weakly self-orthogonal designs invariant under an action of $M_{11}$

Abstract

In this paper we generalize the construction of binary self-orthogonal codes obtained from weakly self-orthogonal designs described by Tonchev in [12] in order to obtain selforthogonal codes over an arbitrary field. We extend construction self-orthogonal codes from orbit matrices of self-orthogonal designs and weakly self-orthogonal 1-designs such that block size is odd and block intersection numbers are even described in [5]. Also, we generalize mentioned construction in order to obtain self-orthogonal codes over an arbitrary field. We construct weakly self-orthogonal designs invariant under an action of Mathieu group M 11 and, from them, binary self-orthogonal codes.Let D be weakly self-orthogonal design and let M be its bˆv incidence matrix. Using suitable extension of M one can obtain self-orthogonal binary code C.Theorem 1 Let D be weakly self-orthogonal design and let M be its b ˆv incidence matrix.1. If D is a self-orthogonal design, than CpDq is a binary self-orthogonal code.2. If D is such that k is even and the block intersection numbers are odd, then the matrix rI b , M, 1s, generates a binary self-orthogonal code.3. If D is such that k is odd and the block intersection numbers are even, then the matrix rI b , M s, generates a binary self-orthogonal code.