Skip to main content
SpringerLink
Log in

Constacyclic codes over the ring \(F_p[u, v]/\langle u^2-1, v^3-v, uv-vu\rangle \) and their applications

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

The objective of this paper is to investigate the structural properties of \((\lambda _1+u\lambda _2+v\lambda _3+v^2\lambda _4+uv\lambda _5+uv^2\lambda _6)\)-constacyclic codes over the ring \(F_p[u, v]/\langle u^2-1, v^3-v, uv-vu\rangle \) for odd prime p. Precisely, we prove that the Gray image of a constacyclic code of length n over this ring is a quasi-constacyclic code of length 6n. Further, some non-trivial examples of constacyclic codes over this ring have been computed. As an application, we have obtained non-binary quantum codes from these classes of constacyclic 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. M. Ashraf, G. Mohammad, Quantum codes from cyclic codes over \(F_3+vF_3\). Int. J. Quantum Inf. 12(6), 1450042 (2014)

    Article  MathSciNet  Google Scholar 

  2. M. Ashraf, G. Mohammad, Construction of quantum codes from cyclic codes over \(F_p+vF_p\). Int. J. Inf. Cod. Theory 3(2), 137–144 (2015)

    MATH  Google Scholar 

  3. M. Ashraf, G. Mohammad, Quantum codes from cyclic codes over \(F_q+uF_q+vF_q+uvF_q\). Quantum Inf. Process. 15(10), 4089–4098 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  4. M. Ashraf, G. Mohammad, Quantum codes over \(F_p\) from cyclic codes over \(F_p[u, v]/\langle u^2-1, v^3-v, uv-vu\rangle \). Cryptogr. Commun. (2018). https://doi.org/10.1007/s12095-018-0299-0

    Article  MATH  Google Scholar 

  5. T. Bag, A.K. Upadhyay, M. Ashraf, G. Mohammad, Quantum codes from cyclic codes over the ring \( F_{p} \left[ u \right]/<u^{3} - u>\). Asian Europ. J. Math. 12(7), 2050008 (2019)

  6. T. Bag, A.K. Upadhyay, Study on negacyclic codes over the ring \(Z_p[u]/<u^{k+1}-u>\). J. Appl. Math. Comput. (2018). https://doi.org/10.1007/s12190-018-1197-5

  7. B. Chen, S. Ling, Zhang: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inform. Theory 61, 1474–1484 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. A. Dertli, Y. Cengellenmis, S. Eren, On quantum codes obtained from cyclic codes over \(A_2\). Int. J. Quantum Inf. 13(3), 1550031 (2015)

    Article  MathSciNet  Google Scholar 

  10. A. Dertli, Y. Cengellenmis, S. Eren, Some results on the linear codes over the finite ring \(F_ 2+ v _1 F_ 2+\dots + v_r F_ 2\). Int. J. Quantum Inf. 14(1), 1650012 (2016)

    Article  MathSciNet  Google Scholar 

  11. J. Gao, Y. Wang, u-Constacyclic codes over \(F_p+uF_p\) and their applications of constructing new non-binary quantum codes. Quantum Inf. Process. 17(4), 1–19 (2018)

    ADS  Google Scholar 

  12. X. Kai, S. Zhu, P. Li, Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inform. Theory 60, 2080–2086 (2014)

    Article  MathSciNet  Google Scholar 

  13. X. Kai, S. Zhu, Quaternary construction of quantum codes from cyclic codes over \(F_4+uF_4\). Int. J. Quantum Inf. 9, 689–700 (2011)

    Article  MathSciNet  Google Scholar 

  14. Y. Liu, R. Li, L. Lv, Y. Ma, A class of constacyclic BCH codes and new quantum codes. Quantum Inf. Process 16(3), 66 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  15. J. Qian, Quantum codes from cyclic codes over \(F_2+vF_2\). J. Inf. Compt. Sci. 10, 1715–1722 (2013)

    Article  Google Scholar 

  16. A.K. Singh, S. Pattanayek, P. Kumar, On Quantum codes from cyclic codes over \(F_2 +uF_2 +u^2F_2\). Asian-Eur. J. Math. 11(1), 1850009 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  19. L. Wang, S. Zhu, New quantum MDS codes derived from constacyclic codes. Quantum Inf. Process. 14, 881–889 (2015)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the learned referee(s) for his/her carefully reading the manuscript and positive comments. The research of second named author is supported by SERB-DST MATRICS Project (Grant No. MTR/2019/000603), India.

Author information

Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Mohammad Ashraf, Shakir Ali & Ghulam Mohammad

Authors
  1. Mohammad Ashraf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shakir Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ghulam Mohammad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shakir Ali.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ashraf, M., Ali, S. & Mohammad, G. Constacyclic codes over the ring \(F_p[u, v]/\langle u^2-1, v^3-v, uv-vu\rangle \) and their applications. Eur. Phys. J. Plus 136, 1215 (2021). https://doi.org/10.1140/epjp/s13360-021-02093-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-02093-5

Search

Navigation