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

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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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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Keywords

  • Double circulant codes
  • Double negacirculant codes
  • Codes over rings
  • Self-dual codes
  • LCD codes
  • Artin conjecture

Mathematics Subject Classification (2010)

  • 94 B15
  • 94 B25
  • 05 E30