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Characterization and enumeration of complementary dual abelian codes

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Abstract

Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary dual in a group algebra \({\mathbb {F}}_{p^\nu }[G]\) has been studied under both the Euclidean and Hermitian inner products, where p is a prime, \(\nu \) is a positive integer and G is an arbitrary finite abelian group. Based on the discrete Fourier transform decomposition for semi-simple group algebras and properties of ideas in local group algebras, the characterization of such codes have been given. Subsequently, the number of complementary dual abelian codes in \({\mathbb {F}}_{p^\nu }[G]\) has been shown to be independent of the Sylow p-subgroup of G and it has been completely determined for every finite abelian group G. In some cases, a simplified formula for the enumeration has been provided as well. The known results for cyclic complementary dual codes can be viewed as corollaries.

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand

    Arunwan Boripan & Patanee Udomkavanich

  2. Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand

    Somphong Jitman

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  1. Arunwan Boripan
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  2. Somphong Jitman
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  3. Patanee Udomkavanich
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Correspondence to Somphong Jitman.

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This research was supported by the Thailand Research Fund under Research Grant MRG6080012.

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Boripan, A., Jitman, S. & Udomkavanich, P. Characterization and enumeration of complementary dual abelian codes. J. Appl. Math. Comput. 58, 527–544 (2018). https://doi.org/10.1007/s12190-017-1155-7

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  • DOI: https://doi.org/10.1007/s12190-017-1155-7

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