Abstract
In recent years, there have been intensive activities in the area of constructing quantum maximum distance separable (MDS for short) codes from constacyclic MDS codes through the Hermitian construction. In this paper, a new class of quantum MDS code is constructed, which extends the result of [Theorems 3.14–3.15, Kai, X., Zhu, S., and Li, P., IEEE Trans. on Inf. Theory, 60(4), 2014, 2080–2086], in the sense that our quantum MDS code has bigger minimum distance.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aly, S. A., Klappenecker, A. and Sarvepalli, P. K., On quantum and classical BCH codes, IEEE Trans. Inf. Theory, 53(3), 2007, 1183–1188.
Ashikhmin, A. and Knill, E., Nonbinary quantum stablizer codes, IEEE Trans. Inf. Theory, 47(7), 2001, 3065–3072.
Calderbank, A. R., Rains, E. M., Shor, P. W. and Sloane, N. J. A., Quantum error correction via codes over GF(4), IEEE Trans. Inf. Theory, 44(4), 1998, 1369–1387.
Chen, H., Some good quantum error-correcting codes from algebraic-geometric codes, IEEE Trans. Inf. Theory, 47(5), 2001, 2059–2061.
Chen, H., Ling, S. and Xing, C., Asymptotically good quantum codes exceeding the Ashikhmin-Litsyn-Tsfasman bound, IEEE Trans. Inf. Theory, 47(5), 2001, 2055–2058.
Chen, H., Ling, S. and Xing, C., Quantum codes from concatenated algebraic-geometric codes, IEEE Trans. Inf. Theory, 51(8), 2005, 2915–2920.
Chen, B., Ling, S. and Zhang, G., Application of constacyclic codes to quantum MDS codes, IEEE Trans. Inf. Theory, 61(3), 2015, 1474–1484.
Feng, K., Quantum codes [[6,2,3]]p and [[7,3,3]]p (p = 3) exist, IEEE Trans. Inf. Theory, 48(8), 2002, 2384–2391.
Feng, K., Ling, S. and Xing, C., Asymptotic bounds on quantum codes from algebraic geometry codes, IEEE Trans. Inf. Theory, 52(3), 2006, 986–991.
Grassl, M., Beth, T. and Rötteler, M., On optimal quantum codes, Int. J. Quantum Inform., 2(1), 2004, 757–766.
Rötteler, M., Grassl, M. and Beth, T., On quantum MDS codes, Information Theory, Proceedings International Symposium on IEEE, 2004, 356.
Guardia, G. G. L., New quantum MDS codes, IEEE Trans. Inf. Theory, 57(8), 2011, 5551–5554.
Hu, X., Zhang, G. and Chen, B., Constructions of new nonbinary quantum codes, Int. J. Theor. Phys., 54(1), 2014, 92–99.
Jin, L., Ling, S., Luo, J. and Xing, C., Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes, IEEE Trans. Inf. Theory, 56(9), 4735–4740, 2010.
Jin, L. and Xing, C., Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes, IEEE Trans. Inf. Theory, 58, 2012, 5484–5489.
Jin, L. and Xing, C., A construction of new quantum MDS codes, IEEE Trans. Inf. Theory, 60, 2014, 2921–2925.
Kai, X. and Zhu, S., New quantum MDS codes from negacyclic codes, IEEE Trans. Inf. Theory, 59(2), 2013, 1193–1197.
Kai, X., Zhu, S. and Li, P., Constacyclic codes and some new quantum MDS codes, IEEE Trans. Inf. Theory, 60(4), 2014, 2080–2086.
Ketkar, A., Klappenecker, A., Kumar, S. and Sarvepalli, P. K., Nonbinary stabilizer codes over finite fields, IEEE Trans. Inf. Theory, 52(11), 2006, 4892–4914.
Knill, E. and Laflamme, R., Theory of quantum error-correcting codes, Phys. Rev. A, 55(2), 1997, 900–911.
Krishna, A. and Sarwate, D. V., Pseudocyclic maximum-distance separable codes, IEEE Trans. Inf. Theory, 36(4), 1990, 880–884.
Li, Z., Xing, L. J. and Wang, X. M., Quantum generalized Reed-Solomon codes: Unified framework for quantum maximum-distance separable codes, Phys. Rev. A, 77, 2008, 012308(1)–012308(4).
Ling, S., Luo, L. and Xing, C., Generalization of Steane’s enlargement construction of quantum codes and applications, IEEE Trans. Inf. Theory, 56(8), 2010, 4080–4084.
Shor, P. W., Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A, 52(4), 1995, 2493–2496.
Steane, A. M., Multiple particle interference and quantum error correction, Proc. Roy. Soc. London A, 452(1), 1996, 2551–2577.
Yang, Y. and Cai, W., On self-dual constacyclic codes over finite fields, Des., Codes Cryptogr., 74(2), 2013, 355–364.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 11171150, 113711138, 11531002) and the Foundation of Science and the Technology on Information Assurance Laboratory (No.KJ-15-009).
Rights and permissions
About this article
Cite this article
Hu, L., Yue, Q. & Zhu, X. New quantum MDS code from constacyclic codes. Chin. Ann. Math. Ser. B 37, 891–898 (2016). https://doi.org/10.1007/s11401-016-1043-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-016-1043-8