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Shi, Minjia; Zhu, Hongwei; Qian, Liqin; Sok, Lin; Solé, Patrick

On self-dual and LCD double circulant and double negacirculant codes over \(\mathbb{F}_q+u\mathbb{F}_q \). (English) Zbl 1446.94195

Cryptogr. Commun. 12, No. 1, 53-70 (2020).
Summary: 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 \(n \rightarrow \infty \). The parameters of examples of modest lengths are computed. Several such codes are optimal.

Cited in 22 Documents

MSC:

94B15 Cyclic codes

Keywords:

double circulant codes; double negacirculant codes; codes over rings; self-dual codes; LCD codes; Artin conjecture

Software:

Code Tables
Cite Review PDF
Full Text: DOI arXiv

References:

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