Skip to main content
SpringerLink
Log in

Two Families of BCH Codes and New Quantum Codes

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In this paper, two families of Hermitian dual-containing Bose-Chau- dhuri-Hocquenghem (BCH) codes with length \(n=a\cdot \frac {q^{2}+ 1}{2}\) and n = b (q2 + 1) are studied, where odd a∣(q − 1) for odd prime power q and b∣ (q + 1) for even prime power q, respectively. Using these non-narrow-sense BCH codes, some new quantum BCH codes are constructed. Most of these quantum codes have better parameters than the ones 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

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, R2493 (1995)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Gottesman, D.: Stabilizer codes and quantum error correction. Ph.D. Thesis, California Institute of Technology (1997)

  4. 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, 1369–1387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Ashikhim, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47, 3065–3072 (2001)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. Grassl, M., Beth, T.: Quantum BCH codes. In: Proc. X. Int. Symp. Theoret. Elec. Eng. Magdeburg, pp. 207–212 (1999)

  10. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. In: Proc. IEEE International Symposium on Information Theory, pp. 1114–1118. Seattle, USA (2006). https://doi.org/10.1109/ISIT.2006.261957

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

    Article  MathSciNet  MATH  Google Scholar 

  12. La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80, 042331 (2009)

    Article  ADS  Google Scholar 

  13. 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 

  14. Li, R., Wang, J., Liu, Y., Guo, G.: New quantum constacyclic codes. Quantum Inf. Processing. (2019). https://doi.org/10.1007/s11128-019-2242-5

  15. Gao, J., Wang, Y.: Quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 57, 682–686 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. La Guardia, G.G.: Quantum Codes Derived from Cyclic Codes. Int. J. Theor. Phys. https://doi.org/10.1007/s10773-017-3399-2 (2018)

  17. Zhang, M., Li, Z., Xing, L., Tang, N.: Construction of some new quantum BCH codes. IEEE Access 6, 36122–36131 (2018). https://doi.org/10.1109/ACCESS.2018.2851075

  18. Gao, J., Wang, Y.: New non-binary quantum codes derived from a class of linear codes. IEEE Access 7, 26418–26421 (2019). https://doi.org/10.1109/ACCESS.2019.2899383

  19. Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Combinat. Designs. 8, 174–188 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Edel, Y.: Table of quantum twisted codes. electronic address. http://www.mathi.uni-heidelberg.de/yves/matritzen/QTBCH/QTBCHIndex.html

  21. Macwilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  23. Li, R., Zuo, F., Liu, Y.: A study of skew symmetric q 2-cyclotomic coset and its application. J. Air Force Eng. Univ. 12(1), 87–89 (2011)

    Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China under Grant Nos.11471011, 11801564 and Natural Science Foundation of Shaanxi under Grant No.2017JQ1032.

Author information

Authors and Affiliations

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

    Junli Wang, Ruihu Li, Yang Liu & Guanmin Guo

Authors
  1. Junli Wang
    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. Guanmin Guo
    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

Wang, J., Li, R., Liu, Y. et al. Two Families of BCH Codes and New Quantum Codes. Int J Theor Phys 58, 2293–2302 (2019). https://doi.org/10.1007/s10773-019-04120-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-019-04120-9

Keywords

Search

Navigation