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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.
Literature
1.
go back to reference Abualrub, T., Siap, I., Aydin, N.: \({{\mathbb{F}}}_{2} {{\mathbb{F}}}_{4}\)-linear cyclic codes. IEEE Trans. Inform. Theory 60(3), 1508–1514 (2014) MathSciNetCrossRef Abualrub, T., Siap, I., Aydin, N.: \({{\mathbb{F}}}_{2} {{\mathbb{F}}}_{4}\)-linear cyclic codes. IEEE Trans. Inform. Theory 60(3), 1508–1514 (2014) MathSciNetCrossRef
2.
go back to reference 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 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
3.
go back to reference 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) 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)
4.
go back to reference 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 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
5.
go back to reference Borges, J., Fernandez-cordoba, C.: \({{\mathbb{F}}}_{2}\) -double cyclic codes. Des. Codes Cryptogrphy 86(3), 463–479 (2018) CrossRef Borges, J., Fernandez-cordoba, C.: \({{\mathbb{F}}}_{2}\) -double cyclic codes. Des. Codes Cryptogrphy 86(3), 463–479 (2018) CrossRef
6.
go back to reference Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (1996) CrossRef Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (1996) CrossRef
7.
go back to reference Cengellenmis, Y., Dougherty, S.T.: Cyclic codes over \( A_k \), In: Proceedings of ACCT2012, Pomorie, Bulgaria Cengellenmis, Y., Dougherty, S.T.: Cyclic codes over \( A_k \), In: Proceedings of ACCT2012, Pomorie, Bulgaria
8.
go back to reference Cengellenmis, Y., Dertli, A., Dougherty, S.T.: Codes over an infinite family of rings with a gray map. Des. Codes Cryptog. 72(3), 559–580 (2014) MathSciNetCrossRef Cengellenmis, Y., Dertli, A., Dougherty, S.T.: Codes over an infinite family of rings with a gray map. Des. Codes Cryptog. 72(3), 559–580 (2014) MathSciNetCrossRef
9.
go back to reference Dougherty, S.T.: Algebraic Coding Theory over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, (2017) Dougherty, S.T.: Algebraic Coding Theory over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, (2017)
10.
go back to reference Dougherty, S.T., Gaborit, P., Harada, M., Munemasa, A., Solé, P.: Type IV self-dual codes over rings. IEEE-IT 45(7), 2345–2360 (1999) MathSciNetCrossRef Dougherty, S.T., Gaborit, P., Harada, M., Munemasa, A., Solé, P.: Type IV self-dual codes over rings. IEEE-IT 45(7), 2345–2360 (1999) MathSciNetCrossRef
12.
go back to reference Huffman, W.C., Pless, V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003) CrossRef Huffman, W.C., Pless, V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003) CrossRef
13.
go back to reference MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977) MATH MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977) MATH
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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