Abstract
A CSS quantum code is succinctly represented as a pair of linear codes \((C_1 ,C_2^{\perp })\) over finite fields \({\mathbb {F}}_{p^e}\) with \(C_2^{\perp }\subset C_1\), where p is a prime and e is a positive integer. In this paper, we present two criteria of the \(C_2^{\perp _s}\subset C_1\) , where \(C_2^{\perp _s}\) denotes the s-Galois dual of \(C_2\) and \(0\le s <e\). Then, using the two criteria, we construct some new quantum codes and a class of new quantum maximum-distance-separable (quantum MDS) codes. In addition, our obtained quantum MDS codes have parameters better than the ones available in the literature.
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Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007)
Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. In: Proceedings of International Symposium on Information Theory (ISIT), pp. 1114–1118 (2006)
Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)
Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over \(GF(4)\). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)
Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78(3), 405–408 (1997)
Calderbank, A.R., Shor, P.W.: Good quantum error correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996)
Chen, B., Ling, S., Zhang, G.: Applications of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015)
Edel, Y.: Some good quantum twisted codes. https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html
Fan, Y., Zhang, L.: Galois self-dual constacyclic codes. Des. Codes Cryptogr. 84, 473–492 (2017)
Hamada, M.: Concatenated quantum codes constructible in polynomial time: efficient decoding and error correction. IEEE Trans. Inf. Theory 54(12), 5689–5704 (2008)
Hurley, T., Hurley, D., Hurley, B.: Entanglement-assisted quantum error-correcting codes from units. arXiv:1806.10875v1 (2018)
Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE Trans. Inf. Theory 60(5), 2921–2925 (2014)
Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2085 (2014)
Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014)
La Guardia, G.G., Palazzo Jr., R.: Constructions of new families of nonbinary CSS codes. Discret. Math. 310, 2935–2945 (2010)
Liu, X., Yu, L., Hu, Peng: New entanglement-assisted quantum codes from \(k\)-Galois dual codes. Finite Field Appl. 55, 21–32 (2019)
Liu, X., Liu, H., Yu, L.: Entanglement-assisted quantum codes from Galois LCD codes. Quantum Inf. Process 19(20), 1–15 (2020)
Ma, Z., Lu, X., Feng, K., Feng, D.: On non-binary quantum BCH codes. In: Lecture Notes in Computer Science, vol. 3959, pp. 675–683 (2006)
Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A. 52(4), 2493–2496 (1995)
Steane, A.M.: Simple quantum error correcting codes. Phys. Rev. A. 54, 4741–4751 (1996)
Steane, A.M.: Enlargement of Calderbank–Shor–Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492–2495 (1999)
Shi, X., Yue, Q., Chang, Y.: Some quantum MDS codes with large minimum distance from generalized Reed-Solomon codes. Cryptogr. Commun. 10, 1165–1182 (2018)
Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf Process. 14, 881–889 (2015)
Acknowledgements
This work was supported by Scientific Research Foundation of Hubei Provincial Education Department of China (Grant No. Q20174503) and the National Science Foundation of Hubei Polytechnic University of China (Grant Nos. 12xjz14A and 17xjz03A).
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Liu, X., Hu, P. New quantum codes from two linear codes. Quantum Inf Process 19, 78 (2020). https://doi.org/10.1007/s11128-020-2575-0
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DOI: https://doi.org/10.1007/s11128-020-2575-0