Cryptography and Communications

, Volume 10, Issue 6, pp 1165–1182 | Cite as

Some quantum MDS codes with large minimum distance from generalized Reed-Solomon codes

  • Xueying Shi
  • Qin YueEmail author
  • Yaotsu Chang


Quantum maximum-distance-separable (MDS) codes are a significant class of quantum codes. In this paper, we mainly utilize classical Hermitian self-orthogonal generalized Reed-Solomon codes to construct five new classes of quantum MDS codes with large minimum distance.


Quantum MDS codes Hermitian self-orthogonal Generalized Reed-Solomon codes 

Mathematics Subject Classification (2010)




This research is supported by National Natural Science Foundation of China (No. 61772015) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17-0225). The paper is also supported by Ministry of Science and Technology, Taiwan, under Grant MOST104-2115-M-214-002-MY2. Additionally, the authors are grateful to the Editor and the anonymous referees for their useful comments and suggestions which helped to improve the presentation of this paper.


  1. 1.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Comb. Des. 8(3), 174–188 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, H., Ling, S., Xing, C.: Quantum codes from concatenated algebraic-geometric codes. IEEE Trans. Inf. Theory 51(8), 2915–2920 (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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(4), 1369–1387 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Feng, K.: Quantum codes [[6, 2, 3]]p and [[7, 3, 3]]p (p ≥ 3) exist. IEEE Trans. Inf. Theory 48(8), 2384–2391 (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. In. J. Quantum Inf. 2(1), 55–64 (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    He, X., Xu, L., Chen, H.: New q-ary quantum MDS codes with distance bigger than \(\frac {q}{2}\). Quantum Inf. Process 15, 2745–2758 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jin, L., Kan, H., Wen, J.: Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes. Des. Codes Cryptogr. 84(3), 463–471 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jin, L., Ling, S., Luo, J., Xing, C.: Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes. IEEE Trans. Inf. Theory 56(9), 4735–4740 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE Trans. Inf. Theory 60(5), 2921–2925 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55(2), 900–911 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inf. Theory 57(8), 5551–5554 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, F., Yue, Q.: New quantum MDS-convolutional codes derived from constacyclic codes. Mod. Phys. Lett. B (2015) in press
  19. 19.
    Li, R., Xu, Z.: Construction of [[n, n − 4, 3]]q quantum codes for odd prime power q. Phys. Rev. A 82(5), 052316(1)–052316(4) (2010)MathSciNetGoogle Scholar
  20. 20.
    Li, Z., Xing, L., Wang, X.: Quantum generalized Reed-Solomon codes: Unified framework for quantum maximum-distance-separable codes. Phys. Rev. A 77 (1), 012308(1)–012308(4) (2008)MathSciNetGoogle Scholar
  21. 21.
    Rötteler, M., Grassl, M., Beth, T.: On quantum MDS codes. In: Proceedings on International Symposium on Information Theory, Chicago, vol. 356 (2004)Google Scholar
  22. 22.
    Shi, X., Yue, Q., Zhu, X.: Construction of some new quantum MDS codes. Finite Fields Appl. 46, 347–362 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Steane, A.M.: Enlargement of Calderbank-Shor-Steane quantum codes. IEEE Trans. Inform. Theory 45(7), 2492–2495 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Process. 14(3), 881–889 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, G., Chen, B.: New quantum MDS codes. Int. J. Quantum Inf. 12(4), 5551–5554 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, T., Ge, G.: Quantum MDS codes with large minimun distance. Des. Codes Cryptogr. 83(3), 503–517 (2016)CrossRefzbMATHGoogle Scholar
  27. 27.
    Zhang, T., Ge, G.: Quantum MDS codes derived from certain classes of polynomials. IEEE Trans. Inf. Theory 62(11), 6638–6643 (2016)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.State Key Laboratory of CryptologyBeijingPeople’s Republic of China
  3. 3.Department of Financial and Computational MathematicsI-SHOU UniversityKaohsiung CityRepublic of China

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