Skip to main content
Log in

Complete classification for simple root cyclic codes over the local ring \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \)

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Let \(R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \). Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over \(\mathbb {Z}_{4}\) and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \).

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

Access this article

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. Abualrub, T., Siap, I.: Cyclic codes over the ring \(\mathbb {Z}_{2}+u\mathbb {Z}_{2}\) and \(\mathbb {Z}_{2}+u\mathbb {Z}_{2}+u^{2}\mathbb {Z}_{2}\). Des. Codes Cryptogr. 42, 273–287 (2007)

    Article  MathSciNet  Google Scholar 

  2. Calderbank, A.R., Hammons, A.R. Jr, Kumar, P.V., Sloane, N.J.A., Solé, P.: A linear construction for certain Kerdock and Preparata codes. Bull. Am. Math. Soc. 29(2), 218–222 (1993)

    Article  MathSciNet  Google Scholar 

  3. Calderbank, A.R., Hammons, A.R. Jr, Kumar, P.V., Sloane, N.J.A., Solé, P.: The \(\mathbb {Z}_{4}\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)

    Article  MathSciNet  Google Scholar 

  4. Cao, Y.: On constacyclic codes over finite chain rings. Finite Fields Appl. 24, 124–135 (2013)

    Article  MathSciNet  Google Scholar 

  5. Cao, Y., Cao, Y., Li, Q.: The concatenated structure of cyclic codes over \(\mathbb {Z}_{p^{2}}\). J. Appl. Math. Comput. 52, 363–385 (2016)

    Article  MathSciNet  Google Scholar 

  6. Cao, Y., Cao, Y., Li, Q.: Concatenated structure of cyclic codes over \(\mathbb {Z}_{4}\) of length 4n. Appl. Algebra in Engrg. Comm. Comput. 10, 279–302 (2016)

    Article  MathSciNet  Google Scholar 

  7. Cao, Y., Li, Q.: Cyclic codes of odd length over \(\mathbb {Z}_{4}[u]/\langle u^{k}\rangle \). Cryptogr. Commun. 9, 599–624 (2017)

    Article  MathSciNet  Google Scholar 

  8. Cao, Y., Cao, Y.: Negacyclic codes over the local ring \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \) of oddly even length and their Gray images. Finite Fields Appl. 52, 67–93 (2018)

    Article  MathSciNet  Google Scholar 

  9. Cao, Y., Gao, Y.: Repeate root cyclic \(\mathbb {F}_{q}\)-linear codes over \(\mathbb {F}_{q^{l}}\). Finite Fields Appl. 31, 202–227 (2015)

    Article  MathSciNet  Google Scholar 

  10. Cao, Y., Cao, Y.: Complete classification for simple root cyclic codes over local rings \(\mathbb {Z}_{p^{s}}[v]/\langle v^{2}-pv\rangle \), https://www.researchgate.net/publication/320620031

  11. Dinh, H.Q.: Complete distance of all negacyclic codes of length 2s over \(\mathbb {Z}_{2^{a}}\). IEEE Trans. Inf. Theory 53, 147–161 (2007)

    Article  Google Scholar 

  12. Dinh, H.Q., Dhompongsa, S., Sriboonchitta, S: On constacyclic codes of length 4ps over \(\mathbb {F}_{p^{m}}+u \mathbb {F}_{p^{m}}\). Discrete Math. 340, 832–849 (2017)

    Article  MathSciNet  Google Scholar 

  13. Dougherty, S.T., Kim, J.-L., Kulosman, H., Liu, H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16, 14–26 (2010)

    Article  MathSciNet  Google Scholar 

  14. Harada, M.: Binary extremal self-dual codes of length 60 and related codes. Des. Codes Cryptogr. 86, 1085–1094 (2018)

    Article  MathSciNet  Google Scholar 

  15. Martínez-Moro, E., Szabo, S., Yildiz, B.: Linear codes over \(\frac {\mathbb {Z}_{4}[x]}{\langle x^{2}+2x\rangle }\). Int. J. Information and Coding Theory 3(1), 78–96 (2015)

    Article  MathSciNet  Google Scholar 

  16. Norton, G., Sălăgean-Mandache, A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra in Engrg. Comm. Comput. 10, 489–506 (2000)

    Article  MathSciNet  Google Scholar 

  17. Shi, M., Zhu, S., Yang, S.: A class of optimal p-ary codes from one-weight codes over Fp[u]/〈um〉. J. Franklin Inst. 350(5), 729–737 (2013)

    Article  Google Scholar 

  18. Shi, M., Qian, L., Sok, L., Aydin, N., Solé, P.: On constacyclic codes over \(\mathbb {Z}_{4}[u]/\langle u^{2}-1\rangle \) and their Gray images. Finite Fields Appl. 45(3), 86–95 (2017)

    Article  MathSciNet  Google Scholar 

  19. Shi, M., Zhang, Y.: Quasi-twisted codes with constacyclic constituent codes. Finite Fields Appl. 39, 159–178 (2016)

    Article  MathSciNet  Google Scholar 

  20. Shi, M., Solé, P., Wu, B.: Cyclic codes and the weight enumerators over \(\mathbb {F}_{2} +v\mathbb {F}_{2} +v^{2}\mathbb {F}_{2}\). Applied and Computational Mathematics 12(2), 247–255 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Wan, Z.-X.: Cyclic codes over Galois rings. Algebra Colloq. 6(3), 291–304 (1999)

    MathSciNet  MATH  Google Scholar 

  22. Wan, Z.-X.: Lectures on finite fields and Galois rings. World Scientific Pub Co Inc (2003)

  23. Wood, J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555–575 (1999)

    Article  MathSciNet  Google Scholar 

  24. Yankov, N., Lee, M.H.: New binary self-dual codes of lengths 50–60. Des. Codes Cryptogr. 73, 983–996 (2014)

    Article  MathSciNet  Google Scholar 

  25. Yildiz, B., Karadeniz, S.: Linear codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\): Macwilliams identities, projections, and formally self-dual codes. Finite Fields Appl. 27, 24–40 (2014)

    Article  MathSciNet  Google Scholar 

  26. Yildiz, B., Aydin, N.: Cyclic codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\) and \(\mathbb {Z}_{4}\) images. International Journal of Information and Coding Theory 2(4), 226–237 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Part of this work was done when Yonglin Cao was visiting the Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11671235, 11801324), the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University)(Grant No. AM201804) and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD09). Yuan Cao and Yonglin Cao contribute equally to this article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yonglin Cao.

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

Cao, Y., Cao, Y. Complete classification for simple root cyclic codes over the local ring \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \). Cryptogr. Commun. 12, 301–319 (2020). https://doi.org/10.1007/s12095-019-00403-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-019-00403-4

Keywords

Mathematics Subject Classification (2010)

Navigation