Skip to main content
SpringerLink
Log in

Quantum codes from linear codes over finite chain rings

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper, we provide two methods of constructing quantum codes from linear codes over finite chain rings. The first one is derived from the Calderbank–Shor–Steane (CSS) construction applied to self-dual codes over finite chain rings. The second construction is derived from the CSS construction applied to Gray images of the linear codes over finite chain ring \({\mathbb {F}}_{p^{2m}}+u{\mathbb {F}}_{p^{2m}}\). The good parameters of quantum codes from cyclic codes over finite chain rings are obtained.

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 computer memory. Phys. Rev. A 52(4), 2493–2496 (1995)

    Article  ADS  Google Scholar 

  2. Steane, A.M.: Simple quantum error correcting codes. Phys. Rev. A 54, 4741–4751 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over \({\mathit{GF}}(4)\). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Qian, J., Ma, W., Gou, W.: Quantum codes from cyclic codes over finite ring. Int. J. Quantum Inf. 7, 1277–1283 (2009)

    Article  MATH  Google Scholar 

  5. Tang, Y., Zhu, S., Kai, X., Ding, J.: New quantum codes from dual-containing cyclic codes over finite rings. Quantum Inf. Process. 15, 4489–4500 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over \({\mathbb{F}}_3 + v{\mathbb{F}}_3\). Int. J. Quantum Inf. 12(1–8), 1450042 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary quantum stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4735–4914 (2006)

    Article  MATH  Google Scholar 

  8. 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 

  9. Liu, X., Liu, H.: LCD codes over finite chain rings. Finite Fields Appl. 34, 1–19 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Norton, G.H., Sǎlǎgean, A.: On the structure of linear and cyclic codes over finite chain rings. AAECC 10, 489–506 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dougherty, S.T., Liu, H.: Independence of vector in codes over rings. Des. Codes Cryptogr. 51, 58–68 (2009)

    Article  MathSciNet  Google Scholar 

  12. Dinh, H., López-Permouth, S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inf. Theory 50(8), 1728–1744 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Edel, Y.: Some good quantum twisted codes. https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html

  14. Dougherty, S.T., Kim, J.L., Kulosman, H., Liu, H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16(1), 14–26 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, B., ling, S., Zhang, G.: Enumeration formulas for self-dual cyclic codes. Finite Fields Appl. 42, 1–22 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to sincerely thank the editor and the anonymous referees for a very meticulous reading of this manuscript and for valuable suggestions which help to create an improved version. This work was supported by Research Funds of Hubei Province, Grant No. D20144401, the Educational Commission of Hubei Province, Grant No. B2015096, and Research Project of Hubei Polytechnic University, Grant No. 17xjzo3A.

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, Hubei, China

    Xiusheng Liu & Hualu Liu

Authors
  1. Xiusheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hualu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiusheng Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Liu, H. Quantum codes from linear codes over finite chain rings. Quantum Inf Process 16, 240 (2017). https://doi.org/10.1007/s11128-017-1695-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-017-1695-7

Keywords

Search

Navigation