On linear codes constructed from finite groups with a trivial Schur multiplier

Using a representation theoretic approach and considering G to be a finite primitive permutation group of degree n with a trivial Schur multiplier, we present a method to determine all binary linear codes of length n that admit G as a permutation automorphism group. In the non-binary case, we can still apply our method, but it will depend on the structure of the stabilizer of a point in the action of G. We show that every binary linear code admitting G as a permutation automorphism group is a submodule of a permutation module defined by a primitive action of G. As an illustration of the method, we consider G to be the sporadic simple group M11 and construct all binary linear codes invariant under G. We also construct some point- and block-primitive 1-designs from the supports of some codewords of the codes in discussion and compute their minimum distances, and in many instances we determine the stabilizers of the non-zero weight codewords.