[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

\({\mathbb {F}}_qR\)-Linear skew cyclic codes

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this work, we introduce the \({\mathbb {F}}_qR\)-linear skew cyclic codes, wh ere \(q=p^s\) is a prime power and \(R={\mathbb {F}}_q+u{\mathbb {F}}_q\) with \(u^2=0\). We provide the algebraic structure of these codes. The dual codes of separable linear skew cyclic codes are also presented. Finally, by using the Gray map from \({\mathbb {F}}_qR\) to \({\mathbb {F}}_q\), we get some optimal linear codes over \({\mathbb {F}}_q\).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abualrub, T., Ghrayeb, A., Aydin, N., Siap, I.: On the construction of skew quasi-cyclic codes. IEEE Trans. Inf. Theory 56(5), 2081–2090 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abualrub, T., Aydin, N., Seneviratne, P.: On \(\theta \)-cyclic codes over \({\mathbb{F}}_{2}+v{\mathbb{F}}_2\). Australas. J. Combin. 54, 115–126 (2012)

    MathSciNet  Google Scholar 

  3. Abualrub, T., Siap, I., Aydin, N.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-additive cyclic codes. IEEE Trans. Inf. Theory 60(3), 1508–1514 (2014)

    Article  MATH  Google Scholar 

  4. Aydogdu, I., Siap, I.: The structure of \({\mathbb{Z}}_{2}{\mathbb{Z}}_{2^s}\)-additive codes: bounds on the minimum distance. Appl. Math. Inf. Sci. 7(6), 2271–2278 (2013)

    Article  MathSciNet  Google Scholar 

  5. Aydogdu, I., Abualrub, T.: The structure of \({\mathbb{Z}}_2{\mathbb{Z}}_2[u]\)-cyclic and constacyclic codes. IEEE Trans. Inf. Theory 63(8), 4883–4893 (2017)

    Article  Google Scholar 

  6. Aydogdu, I., Abualrub, T., Siap, I.: On \({\mathbb{Z}}_2{\mathbb{Z}}_2[u]\)-additive codes. Int. J. Comput. Math. 92, 1806–1814 (2015)

    Article  MathSciNet  Google Scholar 

  7. Aksoy, R., Çalişkan, F.: Self-dual codes over \({\mathbb{F}}_2\times ({\mathbb{F}}_2+v{\mathbb{F}}_2)\). Crypto. Commun. 13(1), 129–141 (2021)

    Article  Google Scholar 

  8. Bhaintwal, M.: Skew quasi-cyclic codes over Galois rings. Des. Codes Cryptogr. 62(1), 85–101 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bag, T., Ashraf, M., Mohammad, G., Upadhyay, A.: Quantum codes from \((1--2u_{1}-2u_{2}-\cdot -2u_{m})\)-skew constacyclic codes over the ring \({\mathbb{F}}_q+u_1{\mathbb{F}}_q+\cdot +u_{2m}{\mathbb{F}}_q\). Quantum Inf. Process. 18(9), 270 (2019)

  10. 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(3), 111737 (2020)

  11. Boucher, D., Geiselmann, W., Ulmmer, F.: Skew cyclic codes. Appl. Algebra Eng. Commun. Comput. 18(4), 379–389 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Boucher, D., Solé, D., Ulmmer, F.: Skew constacyclic codes over Galois rings. Adv. Math. Commun. 2, 273–292 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Borges, J., Fernández-Córdoba, C., Pujol, J., Rif̀a, J., Villanueva, M.: \({\mathbb{Z}}_{2}{\mathbb{Z}}_{4}\)-linear codes: generator matrices and duality. Designs Codes Cryptogr. 54 (2), 167-179 (2010)

  14. Borges, J., Fernández-Córdoba, C., Ten-Valls, R.: On \({\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}\)-additive cyclic codes. Adv. Math. Commun. 12(1), 169–179 (2018)

    Article  MathSciNet  Google Scholar 

  15. Çalişkan, F., Aksoy, R.: Linear codes over \({\mathbb{F}}_2\times ({\mathbb{F}}_2+v{\mathbb{F}}_2)\) and the MacWilliams identities. Appl. Algebra Eng. Commun. Comput. 31(2), 135–147 (2019)

    Article  Google Scholar 

  16. Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Research Reports, 10. Ann Arbor, MI, USA: Historical Jrl. (1973)

  17. Diao, L., Gao, J.: \({\mathbb{Z}}_{p}{\mathbb{Z}}_{p}[u]\)-additive cyclic codes. Int. J. Inf. Coding Theory 5(1), 1–17 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Diao, L., Gao, J., Lu, J.: On \({\mathbb{Z}}_{p}{\mathbb{Z}}_{p}[v]\)-additive cyclic codes. Adv. Math. Commun. 14(4), 555–572 (2020)

    Article  MathSciNet  Google Scholar 

  19. Dinh, H.Q., Pathak, S., Bag, T., Upadhyay, K., Chinnakum, W.: A study of \({\mathbb{F}}_{q}R\)-cyclic codes and their applications in constructing quantum codes. IEEE Access https://doi.org/10.1109/ACCESS.2020.3.32078 (2020)

  20. Gao, J., Shen, L., Fu, F.-W.: A Chinese remainder theorem approach to skew generalized quasi-cyclic codes. Crypto. Commun. 8(1), 51–56 (2016)

    Article  MATH  Google Scholar 

  21. Gao, J., Ma, F., Fu, F.-W.: Skew constacyclic codes over \({\mathbb{F}}_q+v{\mathbb{F}}_q\). Appl. Comput. Math. 6(3), 286–295 (2017)

    Google Scholar 

  22. Gao, J.: Skew cyclic codes over \({\mathbb{F}}_p+v{\mathbb{F}}_p\). J. Appl. Math. Inform. 31(3–4), 337–342 (2013)

    Article  MathSciNet  Google Scholar 

  23. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. Available online at http://www.codetables.de/(1995)

  24. Gursoy, F., Siap, I., Yildiz, B.: Construction of skew cyclic codes over \({\mathbb{F}}_{q}+v{\mathbb{F}}_{q}\). Adv. Math. Commun. 8(3), 313–322 (2014)

    Article  MathSciNet  Google Scholar 

  25. Hou, X., Gao, J.: \({\mathbb{Z}}_{p}{\mathbb{Z}}_{p}[v]\)-additive cyclic codes are asymptotically good. J. Appl. Math. Comput. https://doi.org/10.1007/s12190-020-01466-w (2020)

  26. Jitman, S., Ling, S., Udomkavanich, P.: Skew constacyclic codes over finite chain rings. Adv. Math. Commun. 6(1), 39–63 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kumar, R., Bhaintwal, M.: A class of constacyclic codes and skew constacyclic codes over \({\mathbb{Z}}_{2^s}+u{\mathbb{Z}}_{2^s}\) and their Gray images. J. Appl. Math. Comput. https://doi.org/10.1007/s12190-020-01425-5 (2020)

  28. Li, J., Gao, J., Fu, F.-W., Ma, F.: \({\mathbb{F}}_{q}R\)-linear skew constacyclic codes and their application of constructing quantum codes. Quantum Inf. Process. 19(7), 193 (2020)

    Article  Google Scholar 

  29. Li, J., Gao, J., Fu, F.-W.: Bounds on covering radius of \({\mathbb{F}}_{2}R\)-linear codes. IEEE Commun. Lett. 25(1), 23–27 (2021)

    Article  Google Scholar 

  30. Melakhessou, A., Aydin, N., Hebbache, Z., Guenda, K.: \({\mathbb{Z}}_{q}({\mathbb{Z}}_{q}+u{\mathbb{Z}}_{q})\)-linear skew constacyclic codes. J. Algebra Comb. Discrete Appl. 7(1), 85–101 (2019)

    MATH  Google Scholar 

  31. Ma, F., Gao, J., Li, J., Fu F.-W.: \(\sigma ,\delta \)-Skew quasi-cyclic codes over the ring \({\mathbb{Z}}_{4}+u{\mathbb{Z}}_{4}\). Crypto. Commun. https://doi.org/10.1007/s12095-020-00467-7 (2021)

  32. Ozen, M., Ozzaim, N.T., Ince, H.: Skew quasi-cyclic codes over \({\mathbb{F}}_{q}+v{\mathbb{F}}_{q}\). J. Algebra Appl. 6(4), 1950077 (2019)

    Article  Google Scholar 

  33. Qian, L., Cao, X.: Bounds and optimal \(q\)-ary codes derived from the \({\mathbb{Z}}_{q}R\)-cyclic codes. IEEE Trans. Inf. Theory 66(2), 923–935 (2019)

    Article  Google Scholar 

  34. Siap, I., Abualrub, I., Aydin, N., Seneviratne, P.: Skew cyclic codes of arbitrary length. Int. J. Inf. Coding Theory 2(1), 10–20 (2011)

    MathSciNet  MATH  Google Scholar 

  35. Sharma, A., Bhaintwal, M.: A class of skew-constacyclic codes over \({\mathbb{Z}}_4+u{\mathbb{Z}}_4\) and their \({\mathbb{Z}}_4\)-images. Int. J. Inf. Coding Theory 4(4), 289–303 (2017)

    MathSciNet  Google Scholar 

  36. Shi, M., Wu, R., Krotov, D.S.: On \({\mathbb{Z}}_{p}{\mathbb{Z}}_{p^k}\)-additive codes and their duality. IEEE Trans. Inf. Theory 65(6), 3841–3847 (2019)

    Article  Google Scholar 

  37. Wu, R., Shi, M.: Some classes of mixed alphabet codes with few weights. IEEE Commun. Lett. (2020). https://doi.org/10.1109/LCOMM.2020.3038876

    Article  Google Scholar 

  38. Yao, T., Zhu, S.: \({\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}\)-additive cyclic codes are asymptotically good. Cryptogr. Commun. 12(2), 253–264 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is supported by the 973 Program of China (Grant No. 2013CB834204), the National Natural Science Foundation of China (Grant No. 61571243, 12071264, 11701336, 11626144 and 11671235), the Fundamental Research Funds for the Central Universities of China.

Author information

Authors and Affiliations

  1. Chern Institute of Mathematics and LPMC, and Tianjin Key Laboratory of Network and Data Security Technology, Nankai University, Tianjin, 300071, China

    Juan Li & Fang-Wei Fu

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255091, Shandong, China

    Jian Gao

Authors
  1. Juan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fang-Wei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Gao.

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

Li, J., Gao, J. & Fu, FW. \({\mathbb {F}}_qR\)-Linear skew cyclic codes. J. Appl. Math. Comput. 68, 1719–1741 (2022). https://doi.org/10.1007/s12190-021-01588-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01588-9

Keywords

Mathematics Subject Classification

Search

Navigation