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Quantum MDS codes from BCH constacyclic codes

  • Liqin HuEmail author
  • Qin Yue
  • Xianmang He
Article

Abstract

One central theme in quantum error correction is to construct quantum codes that have large minimum distances. It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. Recently, some quantum MDS codes have been constructed from constacyclic codes. Under these constructions, one of the most important problems is to ensure these constacyclic codes are Hermitian dual-containing. This paper presents a method for determining the maximal designed distance of \([[n, k, d]]_q\) quantum MDS codes from constacyclic codes with fixed n and q. From the method, we can get not only those known quantum MDS codes from constacyclic codes but also a new class of quantum MDS code from Hermitian dual-containing MDS constacyclic code.

Keywords

Quantum MDS code Constacyclic code Cyclotomic coset BCH bound 

References

  1. 1.
    Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cohen, G., Encheva, S., Litsyn, S.: On binary constructions of quantum codes. IEEE Trans. Inf. Theory 45(7), 2495–2498 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, B., Fan, Y., Lin, L., Liu, H.: Constacyclic codes over finite fields. Finite Fields Appl. 18, 1217–1231 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, H., Ling, S., Xing, C.: Quantum codes from concatenated algebraic-geometric codes. IEEE Trans. Inf. Theory 51(8), 2915–2920 (2005)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Feng, K.: Quantum codes \([[6, 2, 3]]_p\) and \([[7, 3, 3]]_p\) (\(p\ge 3\)) exist. IEEE Trans. Inf. Theory 48(8), 2384–2391 (2002)CrossRefGoogle Scholar
  8. 8.
    Feng, K., Ling, S., Xing, C.: Asymptotic bounds on quantum codes from algebraic geometry codes. IEEE Trans. Inf. Theory 52(3), 986–991 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    He, X., Xu, L., Chen, H.: New \(q\)-ary quantum MDS codes with distances bigger than \(\frac{q}{2}\). Quantum Inf. Process. 15(7), 2745–2758 (2016)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jin, L., Xing, C.: Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes. IEEE Trans. Inf. Theory 58, 5484–5489 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE Trans. Inf. Theory 60, 2921–2925 (2014)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55(2), 900–911 (1997)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inf. Theory 57(8), 5551–5554 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ling, S., Luo, L., Xing, C.: Generalization of Steanes enlargement construction of quantum codes and applications. IEEE Trans. Inf. Theory 56(8), 4080–4084 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, S., Xiong, M., Ge, G.: Pseudo-cyclic codes and the construction of quantum MDS codes. IEEE Trans. Inf. Theory 62(4), 1703–1710 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mi, J., Cao, X., Xu, S., Luo, G.: Quantum codes from Hermitian dual-containing cyclic codes. Int. J. Comput. Math. Comput. Syst. Theory 2(3), 97–109 (2017)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Shi, X., Yue, Q., Zhu, X.: Construction of some new quantum MDS codes. Finite Fields Appl. 46, 347–362 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yang, Y., Cai, W.: On self-dual constacyclic codes over finite fields. Des. Codes Cryptogr. 74(2), 355–364 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhang, G., Chen, B., Li, L.: A construction of MDS quantum convolutional codes. Int. J. Theor. Phys. 54, 3182C3194 (2015)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of CyberspaceHangzhou Dianzi UniversityHangzhouPeople’s Republic of China
  2. 2.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  3. 3.Dongguan University of TechnologyGuangdongPeople’s Republic of China

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