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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Abualrub, T., Siap, I., Aydin, N.: \({{\mathbb{F}}}_{2} {{\mathbb{F}}}_{4}\)-linear cyclic codes. IEEE Trans. Inform. Theory 60(3), 1508–1514 (2014)
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)
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., 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)
Borges, J., Fernandez-cordoba, C.: \({{\mathbb{F}}}_{2}\) -double cyclic codes. Des. Codes Cryptogrphy 86(3), 463–479 (2018)
Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (1996)
Cengellenmis, Y., Dougherty, S.T.: Cyclic codes over \( A_k \), In: Proceedings of ACCT2012, Pomorie, Bulgaria
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)
Dougherty, S.T.: Algebraic Coding Theory over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, (2017)
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)
Grassl, M.: Table of Bounds on Linear Codes. http://www.codetables.de/
Huffman, W.C., Pless, V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003)
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977)
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Karbaski, A.S., Abualrub, T. & Dougherty, S.T. Double quadratic residue codes and self-dual double cyclic codes. AAECC 33, 91–115 (2022). https://doi.org/10.1007/s00200-020-00437-9
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DOI: https://doi.org/10.1007/s00200-020-00437-9