Skip to main content
SpringerLink
Log in

A family of negacyclic BCH codes of length \(n=\frac {q^{2m}-1}{2}\)

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

In this paper, we investigate a family of q2-ary narrow-sense and non-narrow-sense negacyclic BCH codes with length \(n=\frac {q^{2m}-1}{2}\), where q is an odd prime power and m ≥ 3 is odd. We propose Hermitian dual-containing conditions for narrow-sense and non-narrow-sense negacyclic BCH codes, and precisely compute the dimensions of these negacyclic BCH codes whose maximal designed distance can achieve \(\delta _{max}^{neg}\). Consequently, many new q-ary quantum codes can be derived from these dual-containing negacyclic BCH codes. Moreover, these new quantum codes are presented either with parameters better than or equal to the ones available in the literature, and also have larger designed distance than those from classical BCH codes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Shor, P.W.: Scheme for reducing decoherence in quantum computing memory. Phys. Rev. A 52(4), R2493–R2496 (1995)

    Article  Google Scholar 

  2. Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  4. Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  MathSciNet  Google Scholar 

  5. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)

    Article  MathSciNet  Google Scholar 

  6. Li, R., Li, X.: Quantum codes constructed from binary cyclic codes. Int. J. Quantum Inf. 2, 265–272 (2004)

    Article  Google Scholar 

  7. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. Proc. Int. Symp. Inf. Theory ISIT, 1114–1118 (2006)

  8. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE. Trans. Inf. Theory 53, 1183–1188 (2007)

    Article  MathSciNet  Google Scholar 

  9. La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 042331(1)-042331 (11) (2009)

    Article  Google Scholar 

  10. Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 12, 0021–0035 (2013)

    MathSciNet  Google Scholar 

  11. Kai, X., Zhu, S., Tang, Y.: Quantum negacyclic codes. Phys. Rev. A 88 (1), 012326(1)-012326 (5) (2013)

    Article  Google Scholar 

  12. Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60, 2080–2086 (2014)

    Article  MathSciNet  Google Scholar 

  13. La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inf. Theory 60(3), 1528–1535 (2014)

    Article  MathSciNet  Google Scholar 

  14. Zhang, T., Ge, G.: Some new classes of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61(9), 5224–5228 (2015)

    Article  MathSciNet  Google Scholar 

  15. Liu, Y., Li, R., Lv, L., Ma, Y.: A class of constacyclic codes and new quantum codes. Quantum Inf. Process. 16, 1–16 (2017)

    Article  MathSciNet  Google Scholar 

  16. Qian, J., Zhang, L.: Improved constructions for nonbinary quantum BCH codes. Int J Theor Phys. 56(4), 1–9 (2017)

    Article  MathSciNet  Google Scholar 

  17. Yuan, J., Zhu, S., Kai, X., Li, P.: On the construction of quantum constacyclic codes. Des. Codes Cryptogr. 85(1), 179–190 (2017)

    Article  MathSciNet  Google Scholar 

  18. Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)

    Article  MathSciNet  Google Scholar 

  19. Aydin, N., Siap, I., Ray-Chaudhuri, D.J.: The structure of 1-generator quasi-twisted codes and new linear codes. Des., Codes Cryptogr. 24, 313–326 (2001)

    Article  MathSciNet  Google Scholar 

  20. Krishna, A., Sarwate, D.V.: Pseudo-cyclic maximum-distance separable codes. IEEE Trans. Inf. Theory 36, 880–884 (1990)

    Article  Google Scholar 

  21. Li, R., Zuo, F., Liu, Y.: A study of skew symmetric q2-cyclotomic coset and its application. J. Air Force Eng. Univ. 12(1), 87–89 (2011)

    Google Scholar 

  22. Lin, X.: Quantum cyclic and constacyclic codes. IEEE Trans. Inf. Theory 50(3), 547–549 (2004)

    Article  MathSciNet  Google Scholar 

  23. Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting codes. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  24. MacWilliams, F.J., Sloane, N.J.A.: The theory of Error-Correcting codes.s North-Holland (1977)

  25. Zhu, S., Sun, Z., Li, P.: A class of negacyclic BCH codes and its application to quantum codes. Des Codes Cryptogr. https://doi.org/10.1007/s10623-017-0441-6 (2017)

  26. Y. Edel’s homepage: http://www.mathi.uni-heidelberg.de/yves/matritzen/QTBCH/QTBCH-Index.html

Download references

Acknowledgments

The authors would like to thank the anonymous referees and the chief editor Prof. Claude Carlet for their very meticulous reading and valuable comments.

Author information

Authors and Affiliations

  1. Department of Basic Sciences, Air Force Engineering University, Xi’an, Shaanxi, 710051, People’s Republic of China

    Guanmin Guo, Ruihu Li, Yang Liu & Junli Wang

Authors
  1. Guanmin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruihu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Junli Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruihu Li.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, G., Li, R., Liu, Y. et al. A family of negacyclic BCH codes of length \(n=\frac {q^{2m}-1}{2}\). Cryptogr. Commun. 12, 187–203 (2020). https://doi.org/10.1007/s12095-019-00387-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-019-00387-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation