SpringerLink
Log in

Improved construction of quantum constacyclic BCH codes

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this work, we investigate a class of narrow-sense constacyclic BCH codes of length \(\frac{q^{2m}-1}{a(q+1)}\) over the finite field \(\mathbb {F}_{q^2}\), where q is a prime power, \(m\ge 2\) is an even integer, and \(a\ne 1\) is a divisor of \(q-1\). The maximum designed distances such that narrow-sense constacyclic BCH codes contain their Hermitian dual codes are determined. The dimensions of the corresponding Hermitian dual-containing codes are worked out. Further, the related quantum codes are constructed. The construction improves the parameters of quantum codes available in the literature.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study

References

  1. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. In: Proceedings of IEEE International Symposium on Information Theory, pp. 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. Ashikhim, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  MATH  Google Scholar 

  4. 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 

  5. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Chuang, L.L., Gershenfeld, N., Kubinec, M.: Experimental implementation of fast quantum searching. Phys. Rev. Lett. 80(15), 3408–3411 (1998)

    Article  ADS  Google Scholar 

  8. Edel, Y.: Some good quantum twisted codes. [Online] https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html. Accessed Apr 2023

  9. Grassl, M., Beth, T.: Quantum BCH codes. In: Proceedings of International Symposium on Theoretical Electrical Engineering, pp. 207-212 (1999)

  10. Grassl, M., Beth, T.: Cyclic quantum error-correcting codes and quantum shift registers. Proc. R. Soc. Lond. A 456(2003), 2689–2706 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Kai, X., Li, P., Zhu, S.: Construction of quantum negacyclic BCH codes. Int. J. Quantum Inf. 16(7), 1850059 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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 

  13. 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 

  14. Ketkar, A., Klappenecker, A., Kumar, S.: Nonbinary stablizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)

    Article  MATH  Google Scholar 

  15. Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lee, J., Lee, E.K., Kim, J., Lee, S.: Quantum shift registers. ar**v:quant-ph/0112107

  17. Li, F., Sun, X.: The Hermitian dual containing non-primitive BCH codes. IEEE Commun. Lett. 25(2), 379–382 (2021)

    Article  ADS  Google Scholar 

  18. Li, R., Wang, J., Liu, Y., Guo, G.: New quantum constacyclic codes. Quantum Inf. Process. 18(5), 127 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 13(1–2), 21–35 (2013)

    MathSciNet  Google Scholar 

  20. Liu, Y., Li, R., Guo, G., Wang, J.: Some nonprimitive BCH codes and related quantum codes. IEEE Trans. Inf. Theory 65(12), 7829–7839 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  23. Rains, E.M.: Non-binary quantum codes. IEEE Trans. Inf. Theory 45(6), 1827–1832 (1999)

    Article  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  25. Song, H., Li, R., Wang, J., Liu, Y.: Two families of BCH codes and new quantum codes. Quantum Inf. Process. 17(10), 270 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Steane, A.M.: Multiple particle interference and quantum error correction. Proc. R. Soc. Lond. A 452(1), 2551–2577 (1996)

    ADS  MathSciNet  MATH  Google Scholar 

  27. Wang, J., Li, R., Liu, Y., Guo, G.: Some negacyclic BCH codes and quantum codes. Quantum Inf. Process. 19(2), 74 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Wang, L., Sun, Z., Zhu, S.: Hermitian dual-containing narrow-sense constacyclic BCH codes and quantum codes. Quantum Inf. Process. 18(10), 323 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Process. 14(3), 881–889 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Wilde, M.M.: Quantum-shift-register circuits. Phys. Rev. A 79(6), 062325(1–16) (2009)

    Article  ADS  MathSciNet  Google Scholar 

  31. 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  MATH  Google Scholar 

  32. Zhang, J., Li, P., Kai, X., Zhu, S.: Some new classes of quantum BCH codes. Quantum Inf. Process. 21(12), 396 (2022)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Zhang, M., Li, Z., **ng, L., Tang, N.: Constructions some new quantum BCH codes. IEEE Access 4, 36122 (2018)

    Article  Google Scholar 

  34. Zhao, X., Li, X., Wang, Q., Yan, T.: A family of Hermitian dual-containing constacyclic codes and related quantum codes. Quantum Inf. Process. 20(5), 186 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Zhu, S., Sun, Z., Li, P.: A class of negacyclic BCH codes and its application to quantum codes. Des. Codes Cryptogr. 86(10), 2139–2165 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This study is supported by the National Natural Science Foundation of China under Grant Nos. 61972126, 62002093, U21A20428 and 12171134)

Author information

Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, Anhui, China

    Ya**g Zhou, **aoshan Kai & Shixin Zhu

Authors
  1. Ya**g Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. **aoshan Kai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shixin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ya**g Zhou.

Ethics declarations

Conflict of interest

All the authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, Y., Kai, X. & Zhu, S. Improved construction of quantum constacyclic BCH codes. Quantum Inf Process 22, 390 (2023). https://doi.org/10.1007/s11128-023-04148-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-023-04148-1

Keywords

Search

Navigation