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On self-dual and LCD double circulant and double negacirculant codes over \(\mathbb {F}_{q}+u\mathbb {F}_{q}\)

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Abstract

Double circulant codes of length 2n over the non-local ring \(R=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,\) are studied when q is an odd prime power, and − 1 is a square in \(\mathbb {F}_{q}\). Double negacirculant codes of length 2n are studied over R when n is even, and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over \(\mathbb {F}_{q}\) are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below for n →∞. The parameters of examples of modest lengths are computed. Several such codes are optimal.

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Acknowledgments

This research is supported by National Natural Science Foundation of China (61672036) and Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China

    Minjia Shi, Hongwei Zhu, Liqin Qian & Lin Sok

  2. Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

    Patrick Solé

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  1. Minjia Shi
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  2. Hongwei Zhu
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  3. Liqin Qian
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  4. Lin Sok
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  5. Patrick Solé
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Correspondence to Minjia Shi.

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Shi, M., Zhu, H., Qian, L. et al. On self-dual and LCD double circulant and double negacirculant codes over \(\mathbb {F}_{q}+u\mathbb {F}_{q}\). Cryptogr. Commun. 12, 53–70 (2020). https://doi.org/10.1007/s12095-019-00363-9

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  • DOI: https://doi.org/10.1007/s12095-019-00363-9

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