Advertisement

International Journal of Theoretical Physics

, Volume 57, Issue 2, pp 453–464 | Cite as

Some Families of Asymmetric Quantum MDS Codes Constructed from Constacyclic Codes

  • Yuanyuan Huang
  • Jianzhang Chen
  • Chunhui Feng
  • Riqing Chen
Article
  • 159 Downloads

Abstract

Quantum maximal-distance-separable (MDS) codes that satisfy quantum Singleton bound with different lengths have been constructed by some researchers. In this paper, seven families of asymmetric quantum MDS codes are constructed by using constacyclic codes. We weaken the case of Hermitian-dual containing codes that can be applied to construct asymmetric quantum MDS codes with parameters \([[n,k,d_{z}/d_{x}]]_{q^{2}}\). These quantum codes are able to correct quantum errors with greater asymmetry. Moreover, these quantum codes constructed in this paper are different from the ones in the literature.

Keywords

Asymmetric quantum codes Constacyclic codes Quantum singleton bound 

Notes

Acknowledgements

We are indebted to anonymous reviewers who have made constructive suggestions for the improvement of this manuscript. The research was supported by the Natural Science Foundation of Fujian Province, China (No.2016J01281,No.2016J01278) and Foundation of Fujian Agriculture and Forestry University (61201406304).

References

  1. 1.
    Steane, A.M.: Simple quantum error-correction codes. Phys. Rev. A 54(6), 4741–4751 (1996)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ashikhmin, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)CrossRefMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 42–331 (2009)CrossRefGoogle Scholar
  6. 6.
    La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inf. Theory 57(8), 5551–5554 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Nielsen, M.A. , Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  8. 8.
    Steane, A.: Enlargement of Calderbank-Shor-Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492–2495 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Aydin, N., Siap, I., Ray-Chaudhuri, D.K.: The structure of 1-generator quasi-twisted codes and new linear codes. Des. Codes Crypt. 24, 313–326 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bakshi, G.K., Raka, M.: A class of constacyclic codes over a finite field. Finite Fields Appl. 18, 362–377 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Berlekamp, E.R.: Negacyclic codes for the Lee metric. Proc. Symp. Combinatorial Math. Appl. 18, 1–27 (1967)CrossRefGoogle Scholar
  12. 12.
    Blackford, T.: Negacyclic duadic codes. Finite Fields Appl. 14, 930–943 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chen, B., Fan, Y., Lin, L., Liu, H.: Constacyclic codes over finite fields. Finite Fields Appl. 18, 1217–1231 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  15. 15.
    Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, New York (1977)MATHGoogle Scholar
  17. 17.
    Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kai, X., Zhu, S., Tang, Y.: Quantum negacyclic codes. Phys. Rev. A 88(88), 18592–18601 (2013)Google Scholar
  19. 19.
    Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Zhang, T., Ge, G.: Some new classes of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61(9), 5224–5228 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    La Guardia, G.G.: On optimal constacyclic codes. Linear Algebra Appl. 496, 594–610 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    La Guardia, G.G.: New families of asymmetric quantum BCH codes. Quantum Inf. Comput. 11(3), 239–252 (2011)MathSciNetMATHGoogle Scholar
  24. 24.
    La Guardia, G.G.: Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes. Quantum Inf. Process 11(2), 591–604 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    La Guardia, G.G. : Asymmetric quantum product codes. International Journal of Quantum Information 10(1), 1250005 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    La Guardia, G.G.: Asymmetric quantum codes: new codes from old. Quantum Inf. Process 12(8), 2771–2790 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    La Guardia, G.G.: On the construction of asymmetric quantum codes. Int. J. Theor. Phys. 53(7), 2312–2322 (2014)MathSciNetMATHGoogle Scholar
  28. 28.
    Leng, R.G., Ma, Z.: Constructions of new families of nonbinary asymmetric quantum BCH codes and subsystem BCH codes. Sci. China Phys. Mech. Astron. 55(3), 465–469 (2012)ADSCrossRefGoogle Scholar
  29. 29.
    Qian, J., Zhang, L.: New optimal asymmetric quantum codes. Mod. Phys. Lett. B 27(2), 1350010 (2013)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Chen, J., Li, J., Lin, J.: New optimal asymmetric quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 53(1), 72–79 (2014)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Xu, G., Li, R., Guo, L., et al.: New optimal asymmetric quantum codes constructed from constacyclic codes. Int. J. Mod. Phys. B 31(5), 1750030 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Chen, J., Li, J., Yang, F., Huang, Y.: Nonbinary Quantum Convolutional Codes Derived from Negacyclic Codes. Int. J. Theor. Phys. 54(1), 198–209 (2015)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Chen, J., Lin, J., Huang, Y.: Asymmetric quantum codes and quantum convolutional codes derived from nonprimitive non-narrow-sense BCH codes. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 98(5), 1130–1135 (2015)ADSCrossRefGoogle Scholar
  34. 34.
    Chen, J., Li, J., Huang, Y., Lin, J.: Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes. Linear Algebra Appl. 475, 186–199 (2015)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ezerman, M.F., Jitman, S., Ling, S., Pasechnik, D.V.: CSS-like constructions of asymmetric quantum codes. IEEE Trans. Inf. Theory 59(10), 6732–6754 (2013)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Ezerman, M.F., Jitman, S., Kiah, H.M., Ling, S.: Pure asymmetric quantum MDS codes from CSS construction: a complete characterization. International Journal of Quantum Information 11(3), 315–328 (2013)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Ezerman, M.F., Jitman, S., Sol, P.: Xing-Ling codes, duals of their subcodes, and good asymmetric quantum codes. Des. Codes Crypt. 75(1), 21–42 (2015)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Li, R., Xu, G., Guo, L.: On two problems of asymmetric quantum codes. Int. J. Mod. Phys. B 28(6), 106–112 (2013)MathSciNetGoogle Scholar
  39. 39.
    Zhang, G., Chen, B., Li, L.: New optimal asymmetric quantum codes from constacyclic codes. Mod. Phys. Lett. B 27(27), 50010 (2014)MathSciNetGoogle Scholar
  40. 40.
    Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes arXiv:1403.2499v1 (2014)
  41. 41.
    Li, F., Yue, Q.: New quantum MDS-convolutional codes derived from constacyclic codes. Mod. Phys. Lett. B 29, 1550252 (2015)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Zhang, G., Chen, B., Li, L.: A construction of MDS quantum convolutional codes. Int. J. Theor. Phys. 54(9), 3182–3194 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Yuanyuan Huang
    • 1
  • Jianzhang Chen
    • 2
  • Chunhui Feng
    • 2
  • Riqing Chen
    • 2
  1. 1.Department of Network EngineeringChengdu University of Information TechnologyChengduChina
  2. 2.College of Computer and Information SciencesFujian Agriculture and Forestry UniversityFuzhouChina

Personalised recommendations