Skip to main content
SpringerLink
Log in

Quantum Codes and Entanglement-Assisted Quantum Codes Derived from One-Generator Quasi-Twisted Codes

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

Abstract

In this paper, we consider a special class of one-generator quasi-twisted (QT) codes of index 2 and explore their algebraic structure. By studying the form of Hermitian dual codes, a sufficient condition for self-orthogonality with Hermitian inner product is proposed. Using some one-generator QT codes, we construct some quantum codes on small finite fields. Moreover, a new method is provided to construct maximal-entanglement entanglement-assisted quantum codes via the class of QT codes. By comparison, our quantum codes and maximal-entanglement entanglement-assisted quantum codes have better parameters than any known 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

References

  1. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Gottesman, D.: Stabilizer Codes and Quantum Error Correction. Ph.D. Thesis, California Institute of Technology (1977)

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

    Article  MATH  Google Scholar 

  5. Ling, S., Luo, J., Xing, C.: Generalization of Stean’s enlargement construction of quantun codes and applications. IEEE Trans. Inf. Theory 56, 4080–4084 (2008)

    Article  MATH  Google Scholar 

  6. Kasami, T.: A Gilber-Vashamov bound for quasi-cyclic codes of rate \(\frac {1}{2}\). IEEE Trans. Inf. Theory 20, 679 (2008)

    Article  Google Scholar 

  7. Chepyzhov, VA.: Gilber-Vashamov bound for quasi-twisted codes of rate \(\frac {1}{n}\). In: Proceedings of the Joint Swedish-Russian International Workshop on Information Theory. Mlle. Sweden, pp. 214–218 (1993)

  8. Siap, I., Aydin, N., Ray-Chaudhuri, D.K.: New ternary quasi-cyclic codes with better minimum distances. IEEE Trans. Inf. Theory 46, 1554–1558 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Aydin, N., Ray-Chaudhuri, D.K.: Quasi-cyclic codes over \(\mathbb {Z}_{4}\) and some new binary codes. IEEE Trans. Inf. Theory 48, 2065–2069 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. CHEN, E: Z.: An explicit construction of 2-generator quasi-twisted codes. IEEE Trans. Inf. Theory 54, 5770–5773 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Daskalov, R., Hristov, P.: New binary one-generator quasi-cyclic codes. IEEE Trans. Inf. Theory 49, 3001–3005 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Daskalov, R., Hristov, P.: New quasi-twisted degenerate ternary linear codes. IEEE Trans. Inf. Theory 49, 2259–2263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. CHEN, E: Z.: New quasi-cyclic codes from simplex codes. IEEE Trans. Inf. Theory 53, 1193–1196 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gao, J., Wang, Y.: Quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 57(3), 682–686 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hou, X., Gao, J.: A minimum distance bound on generalized quasi-twisted codes. Finite Fields Appl. 67, 101712 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Galindo, C., Hernando, F., Mastsumoto, R.: Quasi-cyclic constructions of quantum. Finite Fields Appl. 52, 261–280 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lv, J., Li, R., Wang, J.: Quantum codes derived from one-generator quasi-cyclic codes with hermitian inner product. Int. J. Theor. Phys. 59, 300–312 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lv, J., Li, R., Fu, Q.: Quantum codes derived from quasi-twisted codes of index 2 with Hermitian inner product. IEICE Trans. Fund. Electron. Commun. Comput. Sci. 10, 1411–1415 (2019)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Bosma, W., Cannon, J., Playoust, C.: The MAGMA algebra system I: The user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lv, J., Li, R., Zhang, X.: New non-binary stabilizer quantum codes derived from quasi-negacyclic codes. CISP, Suzhou, China, pp. 1411–1415 (2019)

  22. Lv, J., Li, R., Wang, J.: An explicit construction of quantum stabilizer codes from quasi-cyclic codes. IEEE Communications Letters 1-1 (2020)

  23. Lv, J, Li, R, Yao, Y: Extended quasi-cyclic constructions of quantum codes and entanglement-assisted quantum codes. arXiv:2009.02543

  24. Edel, Y.: Table of quantum twisted codes. Electronic address: www.mathi.uniheidelberg.de/yves/matritzen/QTBCH/QTBCHIndex.html

  25. Bowen, G.: Entanglement required in achieving entanglement-assisted channel capacities. Phys. Rev. A 66, 052313 (2002)

    Article  ADS  Google Scholar 

  26. 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  MATH  Google Scholar 

  27. Lai, C.Y., Brun, T.A.: Entanglement-assisted quantum error correcting codes with imperfect ebits. Phys. Rev. A 86, 032319 (2012)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Lu, L., Li, R., Ma, W., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance. Finite Fields Appl. 53, 309–325 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lu, L., Li, R., Guo, L., Fu, Q.: Maximal entanglement entanglement-assisted quantum codes constructed from linear codes. Quantum Inf. Process. 14 (1), 165–182 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, J., Li, R., Lv, J., Guo, G., Liu, Y.: Entanglement-assisted quantum error correction codes with length n = q2 + 1. Quantum Inf. Process. 18, 1–21 (2019)

    Article  Google Scholar 

  34. Wilde, M.M., Hsieh, M.H., Babar, Z.: Entanglement-assisted quantum turbo codes. IEEE Trans. Inf. Theory 60, 1203–1222 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Grassl, M.: Codetables. electronic address: http://www.codetables.de/ (Accessed 27 Dec. 2019)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Lv, J., Li, R., Wang, J.: New binary quantum codes derived from one-generator quasi-cyclic codes. IEEE Access 7, 85782–85785 (2019)

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by the Graduate Scientific Research Foundation of Department of Basic Sciences of Air Force Engineering University and National Natural Science Foundation of China (Nos.11801564, 11901579).

Author information

Authors and Affiliations

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

    Yu Yao, Yuena Ma & Jingjie Lv

Authors
  1. Yu Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuena Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jingjie Lv
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuena Ma.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yao, Y., Ma, Y. & Lv, J. Quantum Codes and Entanglement-Assisted Quantum Codes Derived from One-Generator Quasi-Twisted Codes. Int J Theor Phys 60, 1077–1089 (2021). https://doi.org/10.1007/s10773-021-04732-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-021-04732-0

Keywords

Search

Navigation