Skip to main content
SpringerLink
Log in

Several families of irreducible constacyclic and cyclic codes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of \([2n, (n-1)/2, d \ge 2(\sqrt{n}+1)]\) irreducible cyclic codes over \({\textrm{GF}}(q)\) and a family of \([(q-1)n, (n-1)/2, d \ge (q-1)(\sqrt{n}+1)]\) irreducible cyclic codes over \({\textrm{GF}}(q)\) are presented, where n is a prime such that \({\textrm{ord}}_{2n}(q)=(n-1)/2\) and \({\textrm{ord}}_{(q-1)n}(q)=(n-1)/2\), respectively. The results in this paper complement earlier works on irreducible constacyclic and cyclic codes over finite fields.

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

Data availability

Not applicable.

Code Availability

Not applicable.

References

  1. Abdukhalikov K., Ho D.: Extended cyclic codes, maximal arcs and ovoids. Des. Codes Cryptogr. 89, 2283–2294 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  2. Berlekamp E.R.: Negacyclic codes for the Lee metric. In: Proceedings of the Conference on Combinatorial Mathematics and Its Applications, pp. 298–316. University of North Carolina Press, Chapel Hill (1968).

  3. Chen B., Fan Y., Lin L., Liu H.: Constacyclic codes over finite fields. Finite Fields Appl. 18(6), 1217–1231 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen B., Dinh H.Q., Fan Y., Ling S.: Polyadic constacyclic codes. IEEE Trans. Inf. Theory 61(9), 4895–4904 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  5. Dahl C., Pedersen J.P.: Cyclic and pseudo-cyclic MDS codes of length \(q+1\). J. Comb. Theory Ser. A 59, 130–133 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  6. Danev D., Dodunekov S., Radkova D.: A family of constacyclic ternary quasi-perfect codes with covering radius 3. Des. Codes Cryptogr. 59, 111–118 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding C., Heng Z.: The subfield codes of ovoid codes. IEEE Trans. Inf. Theory 65(8), 4715–4729 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding C., Tang C.: Designs from Linear Codes, 2nd edn World Scientific, Singapore (2022).

    Book  MATH  Google Scholar 

  9. Ding C., Yang J.: Hamming weights in irreducible cyclic codes. Discr. Math. 313(4), 434–446 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  10. Dong X., Yin S.: The trace representation of \(\lambda \)-constacyclic codes over \({\mathbb{F} }_{q}\). J. Liaoning Normal Univ. (Nat. Sci. Ed.) 33, 129–131 (2010).

    Google Scholar 

  11. Fang W., Wen J., Fu F.: A \(q\)-polynomial approach to constacyclic codes. Finite Fields Appl. 47, 161–182 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  12. Georgiades J.: Cyclic \((q+1, k)\)-codes of odd order \(q\) and even dimension \(k\) are not optimal. Atti Sent. Mat. Fis. Univ. Modena 30(2), 284–285 (1983).

    MathSciNet  MATH  Google Scholar 

  13. Heng Z., Ding C.: A construction of \(q\)-ary linear codes with irreducible cyclic codes. Des. Codes Cryptogr. 87, 1087–1108 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  14. Huffman W.C., Pless V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003).

    Book  MATH  Google Scholar 

  15. Kølve T.: Codes for Error Detection. World Scientfic, Singapore (2007).

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Li F., Yue Q.: The primitive idempotents and weight distributions of irreducible constacyclic codes. Des. Codes Cryptogr. 86, 771–784 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  18. Li F., Yue Q., Liu F.: The weight distribution of constacyclic codes. Adv. Math. Commun. 11(3), 471–480 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  19. Lidl R., Niederreiter H.: Finite Fields. Addison-Wesly, New York (1983).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Mi J., Cao X.: Constructing MDS Galois self-dual constacyclic codes over finite fields. Discr. Math. 344(6), 112388 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  22. Myerson G.: Period polynomials and Gauss sums for finite fields. Acta Arith. 39(3), 251–264 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  23. Pedersen J.P., Dahl C.: Classification of pseudo-cyclic MDS codes. IEEE Trans. Inf. Theory 37(2), 365–370 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  24. Peterson W.W., Weldon E.J. Jr.: Error-Correcting Codes, 2nd edn MIT Press, Cambridge (1972).

    MATH  Google Scholar 

  25. Sharma A., Rani S.: Trace description and Hamming weights of irreducible constacyclic codes. Adv. Math. Commun. 12(1), 123–141 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  26. Shi Z., Fu F.: The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes. Cryptogr. Commun. 12, 29–52 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  27. Singh M.: Weight distributions of all irreducible \(\mu \)-constacyclic codes of length \(\ell ^n\). https://arxiv.org/abs/1806.10600v1.

  28. Sun Z., Zhu S., Wang L.: A class of constacyclic BCH codes. Cryptogr. Commun. 12, 265–284 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang L., Sun Z., Zhu S.: Hermitian dual-containing narrow-sense constacyclic BCH codes and quantum codes. Quantum Inf. Process. 18, 323 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  30. Wolfmann J.: Projective two-weight irreducible cyclic and constacyclic codes. Finite Fields Appl. 14(2), 351–360 (2008).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to Prof. Anuradha Sharma for providing reference [25] and the reviewers as well as the editors for their comments and suggestions that improved the presentation of the paper. All the code examples in this paper were computed with the Magma software package.

Funding

Z. Sun’s research was supported by the National Natural Science Foundation of China under Grant Number 62002093. X. Wang’s research was supported by the National Natural Science Foundation of China under Grant Number 12001175. C. Ding’s research was supported by the Hong Kong Research Grants Council, Proj. No. 16301522.

Author information

Author notes

  1. Xiaoqiang Wang and Cunsheng Ding have contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China

    Zhonghua Sun

  2. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, Hubei, China

    Xiaoqiang Wang

  3. Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    Cunsheng Ding

Authors
  1. Zhonghua Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoqiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cunsheng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhonghua Sun.

Ethics declarations

Conflict of interest

The authors declare there are no conflict of interest.

Additional information

Communicated by G. Ge.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, Z., Wang, X. & Ding, C. Several families of irreducible constacyclic and cyclic codes. Des. Codes Cryptogr. 91, 2821–2843 (2023). https://doi.org/10.1007/s10623-023-01242-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-023-01242-4

Keywords

Mathematics Subject Classification

Search

Navigation