Skip to main content
SpringerLink
Log in

New asymmetric quantum codes over \(\textbf{F}_{q^2}\)

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Two families of asymmetric quantum codes of length \(n=q^{2m}-1\) over \(\textbf{F}_{q^{2}}\) are constructed in this paper. By a detailed analysis of properties about \(q^{2}\)-ary cyclotomic cosets modulo n, Hermitian dual-containing conditions for a family of primitive narrow-sense BCH codes are presented. Consequently, a series of asymmetric quantum BCH codes are constructed via the CSS-like construction and pairs of nested BCH codes. The parameters of new asymmetric quantum codes presented here are better than those available in the literatures before, and the real Z-distance are much larger than \(\delta _{max}+1\).

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

Data availibility

The authors confirm that the data supporting the findings of this study are available within the article.

References

  1. Simple quantum error-correction codes: P. Steane. Phys. Rev. A 64, 4741–4751 (1996)

  2. 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 

  3. Evans, Z.W.E., Stephens, A.M., Cole, J.H., Hollenberg, L.C.L.: Error correction optimisation in the presence of x/z asymmetry, [Online]. e-print arXiv:quant-ph/0709.3875 (2007)

  4. Ioffe, L., Mezard, M.M.: Asymmetric quantum error-correcting codes. Phys. Rev. A 75, 032345 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  5. Aly, S.A.: Quantum error control codes, Ph.D. dissertation, Dept. Comput. Sci., Texas A & M Univ., College Station, TX (2008)

  6. Aly, S.A.: Asymmetric quantum BCH codes. International Conference on Computer Engineering and Systems, 157–162 (2008)

  7. Aly, S.A.: Asymmetric and Symmetric Subsystem BCH Codes and Beyond. [Online]. e-print arXiv:quant-ph/0803.0764 (2008)

  8. Sarvepalli, P.K., Klappenecker, A., Rotteler, M.: Asymmetric quantum codes: constructions, bounds and performance. In: Proceedings of the Royal Society A, 1645C1672 (2009)

  9. Wang, L., Feng, K., Ling, S., Xing, C.: Asymmetric quantum codes: characterization and constructions. IEEE. Trans. Inf. Theory 56, 2938–2945 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ezerman, M.F., Ling, S., Sole, P.: Additive asymmetric quantum codes. IEEE. Trans. Inf. Theory 57, 5536–5550 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ezerman, M.F., Ling, S., Pasechnik, D.V.: CSS-like constructions of asymmetric quantum codes. IEEE Trans. Inf. Theory 59(10), 6732–6754 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. La Guardia, G.G.: New symmetric and asymmetric quantum codes. Quantum Inf. Comp. 11, 0239–0252 (2011)

    Google Scholar 

  13. La Guardia, G.G.: Asymmetric quantum codes: new codes from old. Quantum Inf. Process. 12, 2771–2797 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. La Guardia, G.G.: On the construction of asymmetric quantum codes. Int. J. Theory Phys. 53, 2312–2322 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Xu, G., Li, R.H., Guo, L.B., Lu, L.: New optimal asymmetric quantum codes constructed from constacyclic codes. Int. J. Mod. Phys. B 31(5), 1750030 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Chen, X.J., Zhu, S.X., Kai, X.S.: Two classes of new optimal asymmetric quantum codes. Int. J. Theory Phys. 57(6), 1829–1838 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, Y.Y., Chen, J.Z., Feng, C.H., Chen, R.Q.: Some families of asymmetric quantum MDS codes constructed from constacyclic codes. Int. J. Theory Phys. 57(2), 453–464 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, J.Z., Chen, Y.Q., Huang, Y.Y., Feng, C.H.: New optimal asymmetric quantum codes and quantum convolutional codes derived from constacyclic codes. Quantum Inf. Process. 18(2), 725–742 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, Z.T., Fang, W.J., Fu, F.W.: On pure MDS asymmetric entanglement-assisted quantum error-correcting codes. [Online]. arXiv: 2107.04166 (2021)

  20. Wang, G.H., Tang, C.M., Wei, W.M.: Some new constructions of optimal asymmetric quantum codes. Quantum Inf. Process. 22, 03831–7 (2023)

    Article  MathSciNet  Google Scholar 

  21. Grassl, M.: New Quantum Codes from CSS Codes. [Online]. e-print arXiv:quant-ph/2208.05353 (2022)

  22. Ma, Y.N., Feng, X.Y., Xu, G.: New asymmetric quantum codes over \(F_{q}\). Quantum Inf. Process. 15(7), 2759–2769 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Ma, Y.N., Feng, X.Y., Liu, Y., Guo, G.M.: A class of special code length asymmetric quantum BCH codes. Acta Electron. Sin. 47(11), 2311–2316 (2019)

    Google Scholar 

  24. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. International Symposium on Information Theory, Seattle, 1105–1108 (2006)

Download references

Acknowledgements

We are indebted to the anonymous reviewers for constructive comments and suggestions on our manuscript, which improve the manuscript significantly. This work is supported by National Natural Science Foundation of China under Grant No. U21A20428, Natural Science Foundation Research Program of Shaanxi under Grant Nos. 2021JM-216, 2021JQ-335, 2022JQ-046 and 2023-JC-QN-0033.

Author information

Author notes

  1. Yang Liu, Liangdong Lu and Guanmin Guo have contributed equally to this work.

Authors and Affiliations

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

    Yuena Ma, Yang Liu, Liangdong Lu & Guanmin Guo

Authors
  1. Yuena Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liangdong Lu
    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 Yuena Ma.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts 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

Ma, Y., Liu, Y., Lu, L. et al. New asymmetric quantum codes over \(\textbf{F}_{q^2}\). Quantum Inf Process 22, 175 (2023). https://doi.org/10.1007/s11128-023-03906-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-023-03906-5

Keywords

Search

Navigation