Abstract:
Using a representation theoretic approach and considering G to be a nite
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 de ned 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 the discussion and compute their minimum
distances, and in many instances we determine the stabilizers of non-zero weight codewords.
