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Published in:
19-11-2019 | Original Paper
On the classification of binary self-dual codes admitting imprimitive rank 3 permutation groups
Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 2/2021
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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.