On the classification of binary self-dual codes admitting imprimitive rank 3 permutation groups

Abstract

One of the questions of current interest in coding theory is the following: given a finite non-solvable permutation group G acting transitively on a set $\Omega$, under what conditions on G are self-dual codes invariant under G existent or nonexistent? In this paper, this problem is investigated under the hypothesis that the group G is an imprimitive rank 3 permutation group. It is proven that if G is an imprimitive rank 3 permutation group acting transitively on the coordinate positions of a self-dual binary code C then G is one of ${\mathrm{M}}_{11}$ of degree 22;${\mathrm{Aut}}({\mathrm{M}}_{12})$ of degree 24; $\mathrm{PSL}(2,q)$ of degree $2(q +1)$ for $q {\equiv 1}{({\mathrm{mod}}\,4)};$ $\mathrm{PSL}(m, q)$ of degree $2\times \frac{q^m-1}{q-1}$ for $m \ge 3$ odd and q an odd prime; $\mathrm{PSL}(m, q)$ of degree $2\times \frac{q^m-1}{q-1}$ for $m \ge 4$ even and q an odd prime, and $\mathrm{PSL}(3,2)$ of degree 14. When combined with a result on the classification of binary self-dual codes invariant under primitive rank 3 permutation groups of almost simple type this yields a result on the non-existence of extremal binary self-dual codes invariant under quasiprimitive rank 3 permutation groups of almost simple type.