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The concatenated structure of quasi-abelian codes

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Abstract

The decomposition of a quasi-abelian code into shorter linear codes over larger alphabets was given in Jitman and Ling (Des Codes Cryptogr 74:511–531, 2015), extending the analogous Chinese remainder decomposition of quasi-cyclic codes (Ling and Solé in IEEE Trans Inf Theory 47:2751–2760, 2001). We give a concatenated decomposition of quasi-abelian codes and show, as in the quasi-cyclic case, that the two decompositions are equivalent. The concatenated decomposition allows us to give a general minimum distance bound for quasi-abelian codes and to construct some optimal codes. Moreover, we show by examples that the minimum distance bound is sharp in some cases. In addition, examples of large strictly quasi-abelian codes of about a half rate are given. The concatenated structure also enables us to conclude that strictly quasi-abelian linear complementary dual codes over any finite field are asymptotically good.

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Acknowledgements

The first author was partially supported by PEPS - Jeunes Chercheur-e-s - 2018. The second author was supported by TÜBİTAK Project no. 114F432. The third author is supported by the Swiss Confederation through the Swiss Goverment Excellence Scholarship no: 2019.0413 and by the Swiss National Science Foundation grant n. 188430.

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Authors and Affiliations

  1. Université Paris 8, Laboratoire de Géométrie, Analyse et Applications, LAGA, Université Sorbonne Paris Nord, CNRS, UMR 7539, 93430, Villetaneuse, France

    Martino Borello

  2. Faculty of Engineering and Natural Sciences, Sabancı University, 34956, Istanbul, Turkey

    Cem Güneri

  3. Institute of Mathematics, University of Zurich, Zurich, Switzerland

    Elif Saçıkara

  4. I2M, Centrale Marseille, CNRS, Aix Marseille University, Marseille, France

    Patrick Solé

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  1. Martino Borello
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  2. Cem Güneri
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  3. Elif Saçıkara
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  4. Patrick Solé
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Correspondence to Elif Saçıkara.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”

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Borello, M., Güneri, C., Saçıkara, E. et al. The concatenated structure of quasi-abelian codes. Des. Codes Cryptogr. 90, 2647–2661 (2022). https://doi.org/10.1007/s10623-021-00921-4

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  • DOI: https://doi.org/10.1007/s10623-021-00921-4

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