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

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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.

## Keywords

Imprimitive rank 3 groups Binary self-dual codes Automorphism groups## Mathematics Subject Classification

20D45 94B05## Notes

### Acknowledgements

This paper was written during the tenure of a Core Fulbright Visiting Scholar Program at Michigan State University. I wish to express my sincere gratitude to the Department of Mathematics at Michigan State University for their hospitality, and to Jonathan Hall for insightful discussions and the observations made during the preparation of this article. I extend my gratitude to Alice Devillers for her diligent and patient responses to my queries on the structure of the quasiprimitive imprimitive rank 3 groups, Michael Giudici for his insights into their paper [7], and the anonymous referee whose extremely valuable comments greatly improved the paper in terms of content and presentation.

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