Skip to main content
SpringerLink
Log in

New QEC and EAQEC codes from repeated-root cyclic codes of length \(10p^s\) over finite fields \(\mathbb {F}_{p^m}\)

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Let p be an odd prime with \(p\ne 5\). In this paper, we first provide the structures of repeated-root cyclic codes of length \(10p^s\) over finite fields \(\mathbb {F}_{p^m}\). We then give two methods of constructing good quantum error-correcting (QEC) codes from repeated-root cyclic codes of length \(10p^s\) over finite fields \(\mathbb {F}_{p^m}\). By means of the dimensions of repeated-root cyclic codes of length \(10p^s\) over finite fields \(\mathbb {F}_{p^m}\), we exhibit an effective manner for constructing new EAQEC codes. We show that the usage of these methods brings us many good QEC and EQAEC codes having these advantages: (1) the parameters of our QEC and EQAEC codes are different from all the previous constructions; (2) for repeated-root cyclic codes, our methods allows for easily calculating the dimensions of QEC and EQAEC codes, and the numbers c of pre-shared maximally entangled states of EAQEC codes.

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 datasets were generated or analyzed during the current study. No datasets were generated or analysed during the current study.

References

  1. 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  Google Scholar 

  2. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. In: Proc. Int. Symp. Inform. Theory, ISIT, 1114–1118 (2006)

  3. Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 53, 3065–3072 (2001)

    Article  MathSciNet  Google Scholar 

  4. Brun, T., Devetak, I., Hsieh, M.-H.: Correcting quantum errors with entanglement. Science 53, 436–439 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  5. Berman, S.D.: Semisimple cyclic and Abelian codes \(II\). Kibernetika (Kiev) 3, 21–30 (1967). ((Russian)) English translation: Cybernetics 3, 17–23 (1967)

  6. 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 53, 1369–1387 (1998)

    Article  MathSciNet  Google Scholar 

  7. Castagnoli, G., Massey, J.L., Schoeller, P.A., Seemann, N.: On repeated-root cyclic codes. IEEE Trans. Inf. Theory 37, 337–342 (1991)

    Article  MathSciNet  Google Scholar 

  8. Chen, X., Zhu, S., Jiang, W.: Cyclic codes and some new entanglemen-assisted quantum MDS codes. Des. Codes Cryptogr. 89, 2533–2551 (2021)

    Article  MathSciNet  Google Scholar 

  9. Dinh, H.Q.: Repeated-root constacyclic codes of length \(2p^s\). Finite Fields Appl. 18, 133–143 (2012)

    Article  MathSciNet  Google Scholar 

  10. Dinh, H.Q.: Struture of repeated-root constacyclic codes of length \(3p^s\) and their duals. Discrete Math. 313, 983–991 (2013)

    Article  MathSciNet  Google Scholar 

  11. Dinh, H.Q.: On repeated-root constacyclic codes of length \(4p^s\). Asian-Eur. J. Math. 1, 1–25 (2010)

    Google Scholar 

  12. Dinh, H.Q., Wang, X., Jirakom, S.: On the Hamming distance of constacyclic codes of length \(5p^s\). IEEE Access 8, 44642–46254 (2020)

    Google Scholar 

  13. Dinh, H.Q.: Repeated-root cyclic codes of length \(6p^s\). AMS Contemp. Math. 609, 69–87 (2014)

    Article  Google Scholar 

  14. Dinh, H.Q., Nguyen, B.T., Yamaka, W.: Quantum MDS and synchronizable codes from cyclic and negacyclic codes of length \(2p^s\). IEEE Access 8, 124608–124623 (2020)

    Article  Google Scholar 

  15. Dinh, H.Q., ElDin, R.T., Nguyen, B.T., Tansuchat, R.: MDS constacyclic codes of prime power lengths over fnite felds and construction of quantum MDS codes. Int. J. Theor. Phys. 59, 3043–3078 (2020)

    Article  Google Scholar 

  16. Dinh, H.Q., Nguyen, B.T., Tansuchat, R.: Quantum MDS and synchronizable codes from cyclic codes of length \(5p^s\) over \(\mathbb{F} _{p^{m}}\). Appl. Algebra Eng. Commun. Comput. 34, 931–964 (2023)

    Article  Google Scholar 

  17. Dastbasteh, R., Lisoněk P.: New quantum codes from self-dual codes over \(\mathbb{F}_4\). arXiv:2211.00891v1

  18. Falkner, G., Kowol, B., Heise, W., Zehendner, E.: On the existence of cyclic optimal codes. Atti Semin. Mat. Fis. Univ. Modena 28, 326–341 (1979)

    MathSciNet  Google Scholar 

  19. Grassl, M.: New quantum codes from CSS codes. arXiv:2208.05353v2

  20. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. Available online at http://www.codetables.de, Accessed 2021-04-19

  21. Galindo, C., Hernando, F., Matsumoto, R., Ruano, D.: Entanglement-assisted quantum error-correcting codes over arbitrary finite fields. Quantum Inf. Process 18, 116 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  22. 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  Google Scholar 

  23. Guenda, K., Jitman, S., Gulliver, T.A.: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr. 86, 121–136 (2018)

    Article  MathSciNet  Google Scholar 

  24. Hu, P., Liu, X.: Quantum error-correcting codes from the quantum construction X. Quantum Inf. Process. 22, 366 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  25. Hu, P., Liu, X.: EAQEC codes from two distinct constacyclic codes. Quantum Inf. Process. 22, 100 (2023)

    Article  ADS  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  27. 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  Google Scholar 

  28. Jin, L., Xing, C.: New MDS self-dual codes from generalized Reed–Solomon codes. IEEE Trans. Inf. Theory 63, 1434–1438 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  31. Liu, X., Liu, H., Yu, L.: Entanglement-assisted quantum codes from matrix-product codes. Quantum Inf. Process. 18, 183 (2019)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  35. Massey, J.L., Costello, D.J., Justesen, J.: Polynomial weights and code constructions. IEEE Trans. Inf. Theory 19, 101–110 (1973)

    Article  MathSciNet  Google Scholar 

  36. Ma, Z., Lu, X., Feng, K., Feng, D.: On non-binary quantum BCH codes. Lecture Notes Comput. Sci. 3959, 675–683 (2006)

    Article  MathSciNet  Google Scholar 

  37. Pereira, F.R.F.: Entanglement-assisted quantum codes from cyclic codes. Entropy 25, 37 (2023). https://doi.org/10.3390/e25010037

    Article  ADS  MathSciNet  Google Scholar 

  38. Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86(7), 1565–1572 (2018)

    Article  MathSciNet  Google Scholar 

  39. Roth, R.M., Seroussi, G.: On cyclic MDS codes of length q over \(GF(q)\). IEEE Trans. Inf. Theory 32, 284–285 (1986)

    Article  MathSciNet  Google Scholar 

  40. Sok, L.: On linear codes with one-dimensional Euclidean hull and their applications to EAQECCs. IEEE Trans. Inf. Theory 68(7), 4329–4343 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  42. Silverman, J.H.: A Friendly Introduction to Number Theory. Brown University, Providence (2011)

    Google Scholar 

  43. van Lint, J.H.: Repeated-root cyclic codes. IEEE Trans. Inform. Theory 37, 343–345 (1991)

    Article  MathSciNet  Google Scholar 

  44. Wang, J., Li, R., Lv, J., Guo, G., Liu, Y.: Entanglement-assisted quantum error correction codes with length \(n = q^2 + 1\). Quantum Inf. Process. 18, 1–21 (2019)

    Article  Google Scholar 

  45. Yang, X., Massey, J.L.: The condition for a cyclic code to have a complementary dual. Discrete Math. 126, 391–393 (1994)

    Article  MathSciNet  Google Scholar 

  46. Zhang, X.: Construction of repeated-root constacyclic codes of length \(8p^s\) over \(\mathbb{F} _{p^m}\). Wuhan Univ. J. Nat. Sci. 20, 001–007 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to sincerely thank the editor and the referees for very meticulous readings of this paper and for valuable suggestions which help us to create an improved version. X. Liu was supported by Research Funds of Hubei Province (Grant No. D20144401 and Q20174503) and Research Project of Hubei Polytechnic University (Grant No. numbers 17xjz03A ).

Author information

Authors and Affiliations

  1. School of Science and Technology, College of Arts and Science of Hubei Normal University, Huangshi, 435109, Hubei, China

    Xiusheng Liu

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

    Peng Hu

Authors
  1. Xiusheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peng Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

XL and PH discussed and came up with the initial idea. XL developed the theory and edited the text. PH supervised the findings of this work. All authors provided critical feedback and helped shape the research, analysis and manuscript.

Corresponding author

Correspondence to Peng Hu.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

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

Liu, X., Hu, P. New QEC and EAQEC codes from repeated-root cyclic codes of length \(10p^s\) over finite fields \(\mathbb {F}_{p^m}\). Quantum Inf Process 23, 164 (2024). https://doi.org/10.1007/s11128-024-04374-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-024-04374-1

Keywords

Mathematics Subject Classification

Search

Navigation