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On bounds for quantum error correcting codes over EJ-integers. (English) Zbl 1444.81013

Korpelainen, Nicholas (ed.), TCDM 2018. Proceedings of the 2nd IMA conference on theoretical and computational discrete mathematics, University of Derby, Derby, UK, September 14–15, 2018. Part 1. Amsterdam: Elsevier. Electron. Notes Discrete Math. 70, 89-94 (2018).
Summary: There are some differences between quantum and classical error corrections [M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cambridge: Cambridge University Press (2002; Zbl 1049.81015), see also the review of the 10th anniversary edition (2010; Zbl 1288.81001)]. Hence, these differences should be considered when a new procedure is performed. In our recent study, we construct new quantum error correcting codes over different mathematical structures. The classical codes over Eisenstein-Jacobi (EJ) integers are mentioned in [K. Huber, “Codes over Eisenstein-Jacobi integers”, Contemp. Math. 168, 165 (1994)]. There is an efficient algorithm for the encoding and decoding procedures of these codes [Huber, loc. cit.]. For coding over two-dimensional signal spaces like QAM signals, block codes over these integers \(p=7, 13, 19, 31, 37, 43, 61, \ldots\) can be useful [X. Dong et al. [IEEE Trans. Inf. Theory 44, 1848–1860 (1998; Zbl 0935.94004)]. Thus, in this study, we introduce quantum error correcting codes over EJ-integers. This type of quantum codes may lead to codes with some new and good parameters.
For the entire collection see [Zbl 1409.68021].

MSC:

81P70 Quantum coding (general)
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References:

[1] Calderbank, R.; Shor, P., Good Quantum Error-Correcting Codes Exist, Physical Review, 54, 1098-1105 (1996)
[2] Dong, X.; Soh, C. B.; Gunawan, E.; Tang, L., Groups of Algebraic Integers used for Coding QAM Signals, Information Theory, IEEE, 44, 1848-1860 (1998) · Zbl 0935.94004
[3] Huber, K., Codes over Eisenstein-Jacobi integers, Contemporary Mathematics, 168, 165 (1994) · Zbl 0810.94037
[4] Nielsen, M. A.; Chuang, I. L., Quantum Computation and Quantum Information (2002), Cambridge University Press: Cambridge University Press Cambridge
[5] Yildiz, E.; Demirkale, F., Quantum codes over Eisenstein-Jacobi integers, IEEEXplore
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