Skip to main content
SpringerLink
Log in

New Binary Quantum Codes from Group Rings and Skew Group Rings

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

Abstract

In this paper, we consider group rings and skew group rings and use them to construct binary quantum error-correcting codes(QECCs, for short). We study three classes of groups: direct product of two cyclic groups, dihedral group, direct product of cyclic group and dihedral group. And several special groups. Using these groups we can get construction matrix of the linear codes over \(\mathbb {F}_4\). Then by computer search, we get 12 new binary QECCs with parameters [[40,16,7]], [[42,14,8]], [[48,20,8]], [[50,24,7]], [[54,34,6]], [[55,21,9]], [[60,38,6]], [[60,32,7]], [[70,42,7]], [[72,50,6]], [[75,47,7]], [[75,39,8]]. They both break the best-known lower bound on minimum distance in the Grassl’s code tables.

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

Algorithm 1
Algorithm 2
Algorithm 3

Similar content being viewed by others

Data Availability Statement

Data supporting the results of this study are available upon request from the corresponding author. The availability of this data is limited and the data was used under the license of this study and is therefore not publicly available. However, data may be obtained upon reasonable request by the author and with the permission of the data provider.

References

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

    Article  MathSciNet  Google Scholar 

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Dougherty, S.T., Gildea, J., Taylor, R.: Group rings, G-codes and constructions of self-dual and formally self-dual codes. Des. Codes Cryptogr. 86, 2115–2138 (2018)

    Article  MathSciNet  Google Scholar 

  5. Dougherty, S.T., Sahinkaya, S., Yildiz, B.: Skew G-codes. J. Algebra Appl. 22(02), 2350053 (2023)

    Article  Google Scholar 

  6. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de. Accessed on 2023-08-20

  7. Guan, C., et al.: New binary quantum codes constructed from quasi-cyclic codes. Int. J. Theor. Phys. 61, 172 (2022)

    Article  MathSciNet  Google Scholar 

  8. Hurley, T.: Group rings and rings of matrices. Int. J. Pure Appl. Math. 31, 319–335 (2006)

    MathSciNet  Google Scholar 

  9. Hungerford, T.W.: Algebra. Springer, New York (1974)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press (1996)

  12. Lv, J., et al.: New binary quantum codes derived from one-generator quasi-cyclic codes. IEEE Access 7, 85782–85785 (2019)

    Article  Google Scholar 

  13. Liu, Y., et al.: Lower bounds for quasi-cyclic codes and new binary quantum codes. Symmetry 15, 643 (2023)

    Article  ADS  Google Scholar 

  14. Lu, L., et al.: Quasi-cyclic Hermitian construction of binary quantum codes. arXiv:2210.04487

  15. Şahinkaya, S., Korban, A., Ustun, D.: Maximal entanglement-assisted quantum error correction codes from the skew group ring \(\mathbb{F} _4\rtimes _{\varphi }G\) by a heuristic search scheme. Quantum Inf Process 21, 156 (2022)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  17. Steane, A.: Multiple-particle interference and quantum error correction. Proc. R. Soc. A Math. Phys. Eng. Sci. 452(1954), 2551–2577 (1996)

  18. Wang, J., et al.: A new method of constructing binary quantum codes from arbitrary quaternary linear codes. IEEE Communications Letters 24, 472–476 (2020)

    Article  Google Scholar 

  19. Yao, Y., et al.: New binary quantum codes derived from quasi-twisted codes with hermitian inner product. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 104-A, 1718–1722 (2021)

  20. Yu, C., Zhu, S.: New quantum codes derived from group rings. Int. J. Theor. Phys. 62, 132 (2023)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to extend our heartfelt thanks to all those who contributed to this paper. This work was supported by the National Natural Science Foundation of China under Grant Nos U21A20428 and 12171134.

Author information

Authors and Affiliations

  1. Hefei University of Technology, School of Mathematics, Hefei, 230009, Anhui, People’s Republic of China

    Cong Yu & Shixin Zhu

Authors
  1. Cong Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shixin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Cong Yu and Shixin Zhu wrote the main manuscript text. All authors reviewed the manuscript.

Corresponding author

Correspondence to Shixin Zhu.

Ethics declarations

Conflict of Interest

We declare that this paper represents original work and has not been submitted, in whole or in part, for publication elsewhere. All sources used in the preparation of this manuscript have been properly cited. The work reported in this article was carried out within the framework of appropriate ethical conduct and any potential conflicts of interest have been disclosed. We accept full responsibility for the content and conclusions of this paper.

Competing interests

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

Yu, C., Zhu, S. New Binary Quantum Codes from Group Rings and Skew Group Rings. Int J Theor Phys 63, 27 (2024). https://doi.org/10.1007/s10773-023-05545-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10773-023-05545-z

Keywords

Search

Navigation