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.