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Applicable Algebra in Engineering, Communication and Computing
Published in: Applicable Algebra in Engineering, Communication and Computing 2/2022

29-05-2020 | Original Paper

Double quadratic residue codes and self-dual double cyclic codes

Authors: Arezoo Soufi Karbaski, Taher Abualrub, Steven T. Dougherty

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 2/2022

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Abstract

In this paper, we introduce double Quadratic Residue Codes (QRC) of length \(n=p+q\) for prime numbers p and q in the ambient space \({{\mathbb {F}}} _{2}^{p}\times {{\mathbb {F}}}_{2}^{q}.\) We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when \(p=q.\) We give the main properties for these codes such as their idempotent generators and their duals. We relate these codes to codes over rings and show how they can be used to construct interesting lattices. As an applications of these codes, we provide examples of self-dual, formally self-dual and optimal double QRC. We also provide examples of best known quantum codes that are derived from double-QRC in this setting.
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Borges, J., Fernandez-Cordoba, C., Ten-Valls, R.: \({\mathbb{ F}}_{2}{{\mathbb{F}}}_{4}\)-linear cyclic codes, generator polynomials and dual codes. IEEE Trans. Inform. Theory 62(11), 6348–6354 (2016)MathSciNetCrossRef
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Borges, J., Fernandez-Cordoba, C., Dougherty, S. T., Ten-Valls, R.: Binary images of \({{\mathbb{F}}}_{2}{{\mathbb{F}}}_{4}\)-linear cyclic codes. arxiv:​1707.​03214v1, (2017)
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Borges, J., Fernandez-Cordoba, C., Pujol, J., Rifa, J., Villanueva, M.: \({{\mathbb{F}}}_{2}{{\mathbb{F}}}_{4}\)-linear codes, generator matrix and duality. Des. Codes Cryptogr. 54(2), 167–179 (2009)CrossRef
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Cengellenmis, Y., Dougherty, S.T.: Cyclic codes over \( A_k \), In: Proceedings of ACCT2012, Pomorie, Bulgaria
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Dougherty, S.T.: Algebraic Coding Theory over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, (2017)
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Metadata
Title
Double quadratic residue codes and self-dual double cyclic codes
Authors
Arezoo Soufi Karbaski
Taher Abualrub
Steven T. Dougherty
Publication date
29-05-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 2/2022
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00437-9

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