Skip to main content
SpringerLink
Log in

New quantum codes from matrix-product codes over small fields

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this paper, we provide methods for constructing Hermitian dual-containing (HDC) matrix-product codes over \(\mathbb {F}_{q^2}\) from some non-singular matrices and a special sequence of HDC codes and determine parameters of obtained matrix-product codes when the input matrix and sequence of HDC codes satisfy some conditions. Then, using some nested HDC BCH codes with lengths \(n=\frac{q^4-1}{a} (a=1 ~\)or\(~ a=q\pm 1)\), we construct some HDC matrix-product codes with lengths \(N=\) 2n or 3n and derive nonbinary quantum codes with length N from these matrix-product codes via Hermitian construction. Four classes of quantum codes over \(\mathbb {F}_{q}\) (\(3\le q\le 5\)) are presented, whose parameters are better than those in the literature. Besides, some of our new quantum codes can exceed the quantum Gilbert-Varshamov (GV) bound.

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. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78, 405–408 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  2. Steane, A.M.: Error correctiong codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  3. Gottesman, D.: Stabilizer codes and quantum error correction. Ph.D. Thesis, California Institute of Technology (1997)

  4. Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55(2), 900–911 (1997)

    Article  ADS  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. Steane, A.M.: Enlargement of Calderbank–Shor–Steane quantum codes. IEEE Trans. Inf. Theory 45, 2492–2495 (1999)

    Article  MathSciNet  Google Scholar 

  7. Rains, E.M.: Non-binary quantum codes. IEEE Trans. Inf. Theory 45, 1827–1832 (1999)

    Article  Google Scholar 

  8. Grassl, M., Beth, T.: Quantum BCH codes. In: Proceedings X. International Symposium Theoretical Electronic Engineering, pp. 207-212. Magdeburg (1999)

  9. Ashikhim, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47, 3065–3072 (2001)

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. In: Proceedings of International Symposium on Information Theory, pp. 1114-1118, ISIT (2006)

  12. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53, 1183–1188 (2007)

    Article  MathSciNet  Google Scholar 

  13. La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 042331-1–042331-11 (2009)

    Article  ADS  Google Scholar 

  14. Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 12, 0021–0035 (2013)

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  16. Özbudak, F., Stichtenoth, S.: Note on Niederreiter-Xing’s propagation rule for linear codes. Appl. Algebra Eng. Commun. Comput. 13, 53–56 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Hernando, F., Lally, K., Ruano, D.: Construction and decoding of matrix-product codes from nested codes. Appl. Algebra Eng. Commun. Comput. 20, 497–507 (2009)

    Article  MathSciNet  Google Scholar 

  18. Hernando, F., Ruano, D.: New linear codes from matrix-product codes with polynomial units. Adv. Math. Commum. 4(3), 363–367 (2010)

    Article  MathSciNet  Google Scholar 

  19. Medeni, M.B., Souidi, E.M.: Construction and bound on the performance of matrix-product codes. Appl. Math. Sci. 5(19), 929–934 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Liu, X., Liu, H.: Matrix-prodcut complementary dual codes. arXiv:1604.03774v1 (2016)

  21. Mankean, T., Jitman, S. Matrix-product constructions for self-orthogonal linear Codes. In: Proceedings of the International Conference on Mathematics, Statistics, and Their Applications, pp. 6-10. ICMSA (2016)

  22. Jitman, S., Mankean, T.: Matrix-product constructions for Hermitian self-orthogonal codes. arXiv:1710.04999v1 (2017)

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

    Article  MathSciNet  Google Scholar 

  24. Liu, X., Dinh, H., Liu, H., Yu, L.: On new quantum codes from matrix-prodcut Codes. Cryptogr. Commun. (2017). https://doi.org/10.1007/s12095-017-0242-9

  25. Zhang, T., Ge, G.: Quantum codes from generalized reed-solomon codes and matrix-product codes. arXiv:1508.00978v1 (2015)

  26. Guo, G., Li, R., Guo, L., Wang, J.: Combinatorial construction of a class of quantum codes. J Air Force Eng Univ (Natl Sci Ed) 19(2), 106–110 (2018). (in Chinese)

    Google Scholar 

  27. Liu, X., Liu, H., Yu, L.: Entanglemnt-assisted quantum from matrix-product codes. (2019). https://doi.org/10.1007/s11128-019-23000-z

  28. Van Asch, B.: Matrix-product codes over finite chain rings. Appl. Algebra Eng. Commun. Comput. 19(1), 39–49 (2008)

    Article  MathSciNet  Google Scholar 

  29. Fan, Y., Ling, S., Liu, H.: Matrix-product codes over finite commutative Frobenius rings. Des. Codes Cryptogr. 71, 201–227 (2014)

    Article  MathSciNet  Google Scholar 

  30. Fan, Y., Ling, S., Liu, H.: Homogeneous weight of matrix-product codes over finite principal ideal rings. Finite Fields Appl. 29, 247–267 (2014)

    Article  MathSciNet  Google Scholar 

  31. Sobhani, R.: Matrix-product structure of repeated-root cyclic codes over finite fields. Finite Fields Appl. 39, 216–232 (2016)

    Article  MathSciNet  Google Scholar 

  32. Cao, Y., Cao, Y., Fu, F.: Matrix-product structure of repeated-root constacyclic codes over finite fields. arXiv:1705.08819v2 (2017)

  33. Macwilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977)

    MATH  Google Scholar 

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

    Book  Google Scholar 

  35. Feng, K., Ma, Z.: A finite Gilbert–Varshamov bound for pure stabilizer quantum codes. IEEE. Trans. Inform. Theory 50(12), 3323–3325 (2004)

    Article  MathSciNet  Google Scholar 

  36. Yves Edel’s homepage: http://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html

  37. Wang, J., Li, R., Ma, Y., Liu, Y.: New quantum codes with length \(n=2(q^4-1)\). Procedia Comput Sci 154, 385–677 (2019)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Basic Sciences, Air Force Engineering University, Xi’an, 710051, Shaanxi, People’s Republic of China

    Hao Song, Ruihu Li, Yang Liu & Guanmin Guo

Authors
  1. Hao Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruihu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guanmin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruihu Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11471011 and 11801564, the Natural Science Foundation of Shaanxi province under Grant No. 2019JM-271, the Natural Science Foundation of Department of Basic Sciences in Air Force Engineering University under Grant No. JK2019105.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Song, H., Li, R., Liu, Y. et al. New quantum codes from matrix-product codes over small fields. Quantum Inf Process 19, 226 (2020). https://doi.org/10.1007/s11128-020-02722-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-020-02722-5

Keywords

Search

Navigation