Self-dual and LCD double circulant codes over a class of non-local rings

Let F q be a finite field of order q = p m , where p is an odd prime. This paper presents the study of self-dual and LCD double circulant codes over a class of finite commutative non-chain rings R q = F q + u F q + u 2 F q + ⋯ + u q - 1 F q where u q = u . Here, the whole contribution is two-folded....

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Bibliographic Details
Published in:Computational & Applied Mathematics Vol. 41; no. 6
Main Authors: Prakash, Om, Yadav, Shikha, Islam, Habibul, Solé, Patrick
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.09.2022
Springer Nature B.V
Springer Verlag
Subjects:
Applications of Mathematics
Applied physics
Computational mathematics
Computational Mathematics and Numerical Analysis
Computer Science
Fields (mathematics)
Information Theory
Mathematical Applications in Computer Science
Mathematical Applications in the Physical Sciences
Mathematics
Mathematics and Statistics
Polynomials
Rings (mathematics)
94B15
Self-dual code
94B35
LCD code
Gray map
94B05
Double circulant code
Quasi-cyclic code
94B60
Quasi-cyclic code Mathematics Subject Classification 94B05
ISSN:2238-3603, 0101-8205, 1807-0302
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Summary:Let F q be a finite field of order q = p m , where p is an odd prime. This paper presents the study of self-dual and LCD double circulant codes over a class of finite commutative non-chain rings R q = F q + u F q + u 2 F q + ⋯ + u q - 1 F q where u q = u . Here, the whole contribution is two-folded. First, we enumerate self-dual and LCD double circulant codes of length 2 n over R q , where n is an odd integer. Then by considering a dual-preserving Gray map ϕ , we show that Gray images of such codes are asymptotically good. Second, we investigate the algebraic structure of 1-generator quasi-cyclic (QC) codes over R q for q = 3 . In that context, we present their generator polynomials along with their minimal generating sets and minimum distance bounds. Here, it is proved that ϕ ( C ) is an sq -QC code of length nq over F q if C is an s -QC code of length n over R q .
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ISSN:2238-3603
0101-8205
1807-0302
DOI:10.1007/s40314-022-01947-7