Skip to main content
SpringerLink
Log in

Entanglement-Assisted Quantum Negacyclic BCH Codes

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

Abstract

The entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and important class of quantum codes. The entanglement-assisted formalism can transform arbitrary classical linear codes into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, by decomposing the defining set of negacyclic BCH codes, we construct a class of new EAQECCs with length \(n=\frac {q^{4m}-1}{q^{2}-1}\).

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Steane, A.M.: Simple quantum error-correcting codes. Phys. Rev. A 54(6), 4741–4751 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  4. 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(4), 1369–1387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 55–64 (2004)

    Article  MATH  Google Scholar 

  6. 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 

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

    Article  ADS  Google Scholar 

  8. 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 

  9. Brun, T., Devetak, I., Hsieh, M.-H.: Correcting quantum errors with entanglement. Science 314(5798), 436–439 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Wilde, M.M., Brun, T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77(1-4), 064302 (2008)

    Article  ADS  Google Scholar 

  11. Lai, C.-Y., Brun, T.A.: Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys. Rev. A 88(1-10), 012320 (2013)

    Article  ADS  Google Scholar 

  12. Lü, L., Li, R., Guo, L., Fu, Q.: Maximal entanglement entanglement-assisted quantum codes constructed from linear codes. Quantum Inf. Process. 14, 165–182 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Hsieh, M.-H., Brun, T.A., Devetak, I.: Entanglement-assisted quantum quasicyclic low-density parity-check codes. Phys. Rev. A 79(1-7), 032340 (2009)

    Article  ADS  Google Scholar 

  16. Fujiwara, Y., Clark, D., Vandendriessche, P., Boeck, M.D., Tonchev, V.D.: Entanglement-assisted quantum low-density parity-check codes. Phys. Rev. A 82 (1-19), 042338 (2010)

    Article  ADS  Google Scholar 

  17. Li, R., Zuo, F., Liu, Y.: A study of skew symmetric q 2-cyclotomic coset and its application, vol. 12. (in Chinese) (2011)

  18. Li, R., Xu, G., Lü, L.: Decomposition of defining sets of BCH codes and its applications, vol. 14. (in Chinese) (2013)

  19. Lü, L., Li, R.: Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes. Int. J. Quantum Inf. 12(3), 1450015(1-14) (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process. 16(1-22), 303 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Lü, L., Li, R., Guo, L., Ma, Y., Liu, Y.: Entanglement-assisted quantum MDS codes from negacyclic codes. Quantum Inf. Process. 69(1-23), 17 (2018)

    MathSciNet  MATH  Google Scholar 

  22. 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  MATH  Google Scholar 

  23. Zhu, S., Sun, Z., Li, P.: A class of negacyclic BCH codes and its application to quantum codes. Des. Codes Cryptogr. 86(10), 2139–2165 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, People’s Republic of China

    Xiaojing Chen, Shixin Zhu & Xiaoshan Kai

Authors
  1. Xiaojing Chen
    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

  3. Xiaoshan Kai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shixin Zhu.

Additional information

Publisher’s Note

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

This research is supported by the National Natural Science Foundation of China (No.61772168; No.61572168) and the Natural Science Foundation of Anhui Province (No.1808085MA15).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Zhu, S. & Kai, X. Entanglement-Assisted Quantum Negacyclic BCH Codes. Int J Theor Phys 58, 1509–1523 (2019). https://doi.org/10.1007/s10773-019-04037-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-019-04037-3

Keywords

Search

Navigation