Skip to main content
SpringerLink
Log in

The dual-containing primitive BCH codes with the maximum designed distance and their applications to quantum codes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Classical Bose–Chaudhuri–Hocquenghem (BCH) codes are an important class of cyclic codes in terms of their error-correcting capability. BCH codes that contain their Euclidean (or Hermitian) dual codes have been widely used in constructing quantum stabilizer codes. In this paper, necessary and sufficient conditions on Euclidean and Hermitian dual-containing primitive BCH codes with the maximum designed distance are shown, and necessary conditions on non-narrow-sense primitive BCH codes that do not contain their Euclidean or Hermitian dual codes are given. The results of Euclidean and Hermitian dual-containing primitive BCH codes in (Aly et al. IEEE Trans Inf Theory 53(3):1183–1188, 2007) are extended. Moreover, the dimensions of some non-narrow-sense primitive BCH codes are determined and some quantum codes are constructed.

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

  2. Aly S.A., Klappenecker A., Sarvepalli P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  3. Berlekamp E.R.: The enumeration of information symbols in BCH codes. Bell Syst. Tech. J. 46(8), 1861–1880 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  4. Charpin P.: On a class of primitive BCH-codes. IEEE Trans. Inf. Theory 36(1), 222–228 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen B., Ling S., Zhang G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  6. Ding C., Du X., Zhou Z.: The Bose and minimum distance of a class of BCH codes. IEEE Trans. Inf. Theory 61(5), 2351–2356 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding C., Fan C., Zhou Z.: The dimension and minimum distance of two classes of primitive BCH codes. Finite Fields Appl. 45, 237–263 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  8. 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).

    Article  MathSciNet  MATH  Google Scholar 

  9. Kai X., Zhu S., Li P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  10. Kasami T., Lin S.: Some results on the minimum weight of BCH codes. IEEE Trans. Inf. Theory 18(6), 824–825 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  11. Li C.: Hermitian LCD codes from cyclic codes. Des. Codes Cryptogr. 86(10), 2261–2278 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  12. Li F., Yue Q.: New quantum MDS-convolutional codes derived from constacyclic codes. Mod. Phys. Lett. B (2015). https://doi.org/10.1142/S0217984915502528

  13. Li R., Zuo F., Liu Y., Xu Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 13(2), 0021–0035 (2013).

    MathSciNet  Google Scholar 

  14. Li S., Li C., Ding C., Liu H.: Two families of LCD BCH codes. IEEE Trans. Inf. Theory 63(9), 5699–5717 (2017).

    MathSciNet  MATH  Google Scholar 

  15. Shi X., Yue Q., Zhu X.: Construction of some new quantum MDS codes. Finite Fields Appl. 46, 347–362 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  16. Steane A.M.: Enlargement of Calderbank-Shor-Steane quantum codes. IEEE. Trans. Inf. Theory 45(7), 2492–2495 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  17. Yan H., Liu H., Li C., Yang S.: Parameters of LCD BCH codes with two lengths. Adv. Math. Commun. 12(3) (2018)

  18. Zhang G., Chen B.: New quantum MDS codes. Int. J. Quantum Inf. 12(4), 5551–5554 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang T., Ge G.: Quantum MDS codes with large minimum distance. Des. Codes Cryptogr. 83(3), 503–517 (2016).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research is supported by National Natural Science Foundation of China (No. 61772015), Postgraduate Research & Practice Innovation Program of Jiangsu Province (Nos. KYCX17_0225 and KYCX18_0241), and the Fundamental Research Funds for the Central Universities (No. NZ2018005). The paper is also supported by Foundation of Science and Technology on Information Assurance Laboratory (No. KJ-17-010), the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant No. 17KJB110018), and Guangxi Natural Science Foundation(2016GXNSFDA380017). 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.

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Xueying Shi, Qin Yue & Yansheng Wu

  2. State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, People’s Republic of China

    Xueying Shi, Qin Yue & Yansheng Wu

Authors
  1. Xueying Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qin Yue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yansheng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xueying Shi.

Additional information

Communicated by J. Bierbrauer.

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

Shi, X., Yue, Q. & Wu, Y. The dual-containing primitive BCH codes with the maximum designed distance and their applications to quantum codes . Des. Codes Cryptogr. 87, 2165–2183 (2019). https://doi.org/10.1007/s10623-019-00610-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-019-00610-3

Keywords

Mathematics Subject Classification

Search

Navigation