Permutation groups and binary self-orthogonal codes

Let G be a permutation group on an n-element set Ω. We study the binary code C (G, Ω) defined as the dual code of the code spanned by the sets of fixed points of involutions of G. We show that any G-invariant self-orthogonal code of length n is contained in C (G, Ω). Many self-orthogonal codes related to sporadic simple groups, including the extended Golay code, are obtained as C (G, Ω). Some new self-dual codes invariant under sporadic almost simple groups are constructed.