Abstract
Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them, generalizing some old results of MacWilliams on the enumeration of circulant orthogonal matrices. These formulae, in turn, are instrumental in deriving a Varshamov–Gilbert bound on the relative minimum distance of this family of codes.
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Acknowledgements
The authors are indebted to Hatoon Shoaib for helpful discussions. The second author was supported by TÜBİTAK 2214—International Doctoral Research Fellowship Programme. All authors thank the anonymous referees for helpful suggestions that greatly improved the presentation and content of the material.
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Communicated by D. Ghinelli.
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Alahmadi, A., Özdemir, F. & Solé, P. On self-dual double circulant codes. Des. Codes Cryptogr. 86, 1257–1265 (2018). https://doi.org/10.1007/s10623-017-0393-x
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DOI: https://doi.org/10.1007/s10623-017-0393-x