Skip to main content
SpringerLink
Log in

QEC and EAQEC codes from cyclic codes over non-chain rings

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Let \(R_q={\mathbb {F}}_q+u{\mathbb {F}}_q+ v{\mathbb {F}}_q+uv{\mathbb {F}}_q\) with \(u^2=u,v^2=v, vu=vu\). In this paper, by using CSS and Steane\('\)s constructions, we construct quantum error-correcting (abbreviated to QEC) codes from the Euclidean sums of cyclic codes of length n over \(R_q\). Then, concrete examples are presented to construct new QSCs. We also construct entanglement-assisted quantum error-correcting codes (abbreviated to EAQEC codes) by means of the Euclidean hulls of cyclic codes of length n over \(R_q\). In addition, our obtained EAQEC codes have parameters better than the ones 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

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 availability

Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

References

  1. Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over \({F }_q+u{F }_q+v{F }_q+uv{F }_q\). Quantum Inf. Process. 15(10), 4089–4098 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  5. Ashraf, M., Mohammad, G.: Quantum codes over \({F }_p[u, v]/\langle u^2-1, v^3-v, uv- vu\rangle \). Cryptogr. Commun. 11, 325–335 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brun, T., Devetak, I., Hsieh, H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Ball, S.: Some constructions of quantum MDS codes. Des. Codes Cryptogr. 73(2), 417–424 (2020)

  8. Edel, Y.: Some good quantum twisted codes. accessed on Jul., [Online] Available: https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html (2020)

  9. Dinh, H.Q., Le, H.T., Nguyen, B.T., Tansuchat, R.: Quantum MDS and synchronizable codes from cyclic and negacyclic codes of length \(4p^s\) over \({F }_{p^m}\). Quantum Inf. Process. 20, 373 (2021)

    Article  MATH  ADS  Google Scholar 

  10. Fang, W., Fu, F.: Some new constructions of quantum MDS codes. IEEE Trans. Inf. Theory 65(12), 7840–7847 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grassl, M., Beth, T., Röttler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 757–775 (2004)

    Article  Google Scholar 

  12. Fan, J., Chen, H., Xu, J.: Construction of \(q\)-ary entanglement-assisted quantum MDS codes with minmum distanc greater than \(q+1\). Quantum Inf. Comput. 16, 0423–0434 (2016)

    MathSciNet  Google Scholar 

  13. Fang, W., Fu, F.W., Li, L., Zhu, S.: Euclidean and Hermitian Hulls of MDS codes and their applications to EAQECCs. EEE Trans. Inf. Theory 60(6), 3527–3537 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Galindo, C., Hernando, F., Matsumoto, R., Ruano, D.: Entanglement-assisted quantum error-correcting codes over arbitrary finite fields. Quantum Inf. Process. 19, 116 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. Guenda, K., Gulliver, T.A., Jitman, S., Thipworawimon, S.: Linear \(l\)-intersection pairs of codes and their applications. Des. Codes Cryptogr. 88, 133–152 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gao, Y., Gao, J., Fu, F.: Quantum codes from cyclic codes over the ring \({F }_q +v_1{F }_q +\ldots + v_r{F }_q\). AAECC 30, 161–174 (2019)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  18. Hamada, M.: Concatenated quantum codes constructible in polynomial time: Efficient decoding and error correction. EEE Trans. Inf. Theory 54, 5689–5704 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu, P., Liu, H.: Three classes of new EAQEC MDS codes. Quantum Inf. Process. 20, 103 (2021)

    Article  MathSciNet  ADS  Google Scholar 

  20. Islam, H., Prakash, O.: New quantum codes from cyclic codes over \({F}_{p}[u,v,w]/\langle u^2-1,v^2-1, uv-vu,vw-wv,wu-uw\rangle \), Journal of Applied Mathematics and Computing, https://doi.org/10.1007/s12190-018-01230-1

  21. Jin, L., Ling, S., Luo, J.: Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes. IEEE Trans. Inf. Theory 56(9), 4735–4740 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE Trans. Inf. Theory 65, 2921–2925 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Luo, L., Ma, Z., Wei, Z., Leng, R.: Non-binary entanglement-assisted quantum stabilizer codes. Sci. China Inf. Sci. 60, 11–14 (2017)

    Article  Google Scholar 

  26. Luo, G., Cao, X.: Two new families of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf. Process. 18, 89 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  27. Liu, H., Liu, X.: Constructions of quantum MDS codes. Quantum Inf. Process. 20, 14 (2021)

    Article  MathSciNet  ADS  Google Scholar 

  28. Liu, X., Yu, L., Hu, P.: New entanglement-assisted quantum codes from \(k\)-Galois dual codes. Finite Fields Appl. 55, 21–32 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, X., Liu, H., Yu, L.: New EAQEC codes constructed from Galois LCD codes. Quantum Inf. Process. 19, 20 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  30. Liu, X., Liu, H.: Quantum codes from linear codes over finite chain rings. Quantum Inf. Process. 16, 240 (2017)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  31. Liu, H., Hu, P.: New quantum codes from two linear codes. Quantum Inf. Process. 19, 78 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  32. Liu, H., Liu, X.: New EAQEC codes from cyclic codes over over \({F }_q+u{F }_q\). Quantum Inf. Process. 19, 85 (2020)

    Article  ADS  Google Scholar 

  33. Ling, S., Xing, C.P.: Coding Theory\(-\)A First Course. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  34. Sok, L., Qian, G.: Linear codes with arbitrary dimensional hull and their applications to EAQECCs. Quantum Inf. Process. 21, 72 (2022)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  35. Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), 2493–2496 (1995)

    Article  ADS  Google Scholar 

  36. Steane, A.M.: Multiple particle interference and quantum error correction. Proc. R. Soc. Lond. A 452, 2551–2557 (1996)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  37. Wilde, M.M., Brun, T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008)

    Article  ADS  Google Scholar 

  38. Yao, T., Shi, M., Sole, P.: Skew cyclic codes over \({F }_q + u{F }_q + v{F }_q + uv{F }_q\). J. Algebra Comb. Discrete Appl. 2(3), 163–168 (2015)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by Research Funds of Hubei Province, Grant No. Q20174503.

Author information

Authors and Affiliations

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

    Xiaoyan Zhang

Authors
  1. Xiaoyan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoyan Zhang.

Ethics declarations

Conflict of interest

The author declare that she have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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

Zhang, X. QEC and EAQEC codes from cyclic codes over non-chain rings. Quantum Inf Process 21, 389 (2022). https://doi.org/10.1007/s11128-022-03734-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-022-03734-z

Keywords

Mathematics Subject Classification

Search

Navigation