Skip to main content
Log in

Hermitian self-orthogonal matrix product codes and their applications to quantum codes

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this paper, we propose a construction of quantum codes from Hermitian self-orthogonal matrix product codes over the finite fields. This construction is applied to obtain numerous new quantum codes, and all of them have higher rate than current quantum codes available.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

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

    Article  MathSciNet  Google Scholar 

  3. Blackmore, T., Norton, G.H.: Matrix-product codes over \(\mathbb{F} _q\). Appl. Algebra Eng. Commun. Comput. 12, 477–500 (2001)

    Article  Google Scholar 

  4. Bag, T., Dinh, H., Upadhyay, A., Bandi, R., Yamaka, W.: Quantum codes from skew constacyclic codes over the ring \(\mathbb{F} _q[u, v]/\langle u^2-1, v^2-1, uv-vu\rangle \). Discrete Math. 343, 111737 (2020)

    Article  MathSciNet  Google Scholar 

  5. 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  Google Scholar 

  6. Cao, M., Wang, H., Cu, J.: Construction of quantum codes from matrix-product codes. IEEE Commun. Lett. 24(4), 706–710 (2020)

    Article  Google Scholar 

  7. Cannon, J., Playoust, C.: An Introduction to Magma. The University of Sydney, Sydney (1994)

    Google Scholar 

  8. Galindo, C., Hernando, F., Ruano, D.: New quantum codes from evaluation and matrix-product codes. Finite Fields Their Appl. 36, 98–120 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE Trans. Inf. Theory 60(5), 2921–2925 (2014)

    Article  MathSciNet  Google Scholar 

  11. La Guardia, G.G.: Quantum Error Correction. Springer, Berlin (2020)

    Book  Google Scholar 

  12. Liu, X.S., Dinh, H.Q., Liu, H., Yu, L.: On new quantum codes from matrix product codes. Cryptogr. Commun. 10(4), 579–589 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  13. Liu, X.S., 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 

  14. Ma, F., Cao, J.: New non-binary quantum codes from constacyclic codes over \(\mathbb{F} _q[u, v]/\langle u^2-1, v^2-v, uv-vu\rangle \). Adv. Math. Commun. 13(3), 421–434 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Sok, L.: New families of quantum stabilizer codes from Hermitian self-orthogonal algebraic geometry codes. arXiv:2110.00769v1

  17. Verma, R.K., Prakash, O., Singh, A., Islam, H.: New quantum codes from skew constacyclic codes. Adv. Math. Commun. (2021). https://doi.org/10.3934/amc.2021028

    Article  Google Scholar 

  18. Zhang, T., Ge, G.: Quantum codes from generalized Reed–Solomon codes and matrix-product codes. arXiv:1508.00978v1

Download references

Acknowledgements

This work was supported by Research Funds of Hubei Province, Grant No. D20144401.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiaoyan Zhang.

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. Hermitian self-orthogonal matrix product codes and their applications to quantum codes. Quantum Inf Process 23, 108 (2024). https://doi.org/10.1007/s11128-024-04314-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-024-04314-z

Keywords

Mathematics Subject Classification

Navigation