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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.

In this paper, we construct, using computations withMagma, a ternary code C from a primitive permutation representation of degree 15 of the group PSL2(9) by Key-Moori Method 1. The code C is an optimal code invariant under the group S6. We consider the action of the automorphism group S6 on C and its dual. By taking the support of any codeword ? of weight l and orbiting it under S6, 1-(15, l, kl) designs are obtained, where kl = l|?S6 |/15. For any codeword, the structure of the stabilizer in S6 is determined and primitivity of S6 on each design is examined. It is shown that the complement of one of these designs is actually a new design D with parameters 2-(15, 7, 36). Moreover, Aut(D) ? S6.

In this paper, we construct, using computations withMagma, a ternary code C from a primitive permutation representation of degree 15 of the group PSL2(9) by Key-Moori Method 1. The code C is an optimal code invariant under the group S6. We consider the action of the automorphism group S6 on C and its dual. By taking the support of any codeword ? of weight l and orbiting it under S6, 1-(15, l, kl) designs are obtained, where kl = l|?S6 |/15. For any codeword, the structure of the stabilizer in S6 is determined and primitivity of S6 on each design is examined. It is shown that the complement of one of these designs is actually a new design D with parameters 2-(15, 7, 36). Moreover, Aut(D) ? S6.