An infinite family of binary cyclic codes and two infinite families of ternary cyclic codes with good parameters (original) (raw)

Abstract

In this paper, two infinite families of ternary cyclic codes with length \( 3^m-1 \) are constructed, where m is odd and m is sufficiently large. The minimum distance of one family of codes satisfies \( d\ge 6(3^{\frac{m-1}{2}}-1)+1 \), and the minimum distance of the other family of codes satisfies \( d\ge 3^{\frac{m-1}{2}}-1 \). The dimensions of these codes are close to half their length when m is sufficiently large. We also give an infinite family of binary cyclic codes with parameters \( [2^m-1,2^{m-1},d\ge 7\times 2^{(m-3)/2}+1]_2 \), where \( m\equiv 1\pmod {4} \) and \( m\ge 23 \). This family of binary cyclic codes has a better lower bound on the minimum distance than that given in Sun (2023).

Access this article

Log in via an institution

Subscribe and save

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Data Availability

No datasets were generated or analysed during the current study.

References

  1. Assmus, E. F., Key, J. D.: Polynomial codes and finite geometries, in Handbook Coding Theory, V. S. Pless and W. C. Huffman, Eds. Amsterdam, Netherlands: Elsevier, pp. 1269-1343 (1998)
  2. Chen, T., Ding, C., Li, C., Sun, Z.: Four infinite families of ternary cyclic codes with a square-root-like lower bound. Finite Fields Appl. 92, 102308 (2023)
  3. Ding, C.: Cyclotomic constructions of cyclic codes with length being the product of two primes. IEEE Trans. Inf. Theory 58(4), 2231–2236 (2012)
    Article MathSciNet MATH Google Scholar
  4. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes, Online available at http://www.codetables.de
  5. Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge Univ. Press, Cambridge, U.K. (2003)
  6. Liu, H., Li, C., Ding, C.: Five infinite families of binary cyclic codes and their related codes with good parameters. Finite Fields Appl. 91, 102270 (2023)
  7. Sun, Z.: Several families of binary cyclic codes with good parameters. Finite Fields Appl. 89, 102200 (2023)
  8. Sun, Z., Li, C., Ding, C.: An infinite family of binary cyclic codes with best parameters. IEEE Trans. Inf. Theory 70(4), 2411–2418 (2024)
  9. Tang, C., Ding, C.: Binary \([n,(n + 1)/2]\) cyclic codes with good minimum distances. IEEE Trans. Inf. Theory 68(12), 7842–7849 (2022)
    Article MathSciNet Google Scholar
  10. Xiong, M.: On cyclic codes of composite length and the minimum distance. IEEE Trans. Inf. Theory 64(9), 6305–6314 (2018)
    Article MathSciNet MATH Google Scholar
  11. Xiong, M., Zhang, A.: On cyclic codes of composite length and the minimum distance II. IEEE Trans. Inf. Theory 67(8), 5097–5103 (2021)
    Article MathSciNet MATH Google Scholar

Download references

Acknowledgements

The authors thank Dr. Zhonghua Sun for helpful discussion.

Funding

This research was supported in part by the National Natural Science Foundation of China under Grants (12471490, 12201170 and 12301663), in part by the Natural Science Foundation of Anhui Province under Grant (2308085QA10) and in part by the China Postdoctoral Science Foundation under Grant (2023M740016).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China
    Haodong Lu
  2. School of Cyberspace Security, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, China
    Liqin Qian
  3. Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China
    Minjia Shi

Authors

  1. Haodong Lu
  2. Liqin Qian
  3. Minjia Shi

Contributions

All authors wrote the main manuscript text and reviewed the manuscript.

Corresponding author

Correspondence toMinjia Shi.

Ethics declarations

Competing Interests

The authors declare no competing interests.

They have no known competing financial or non-financial interests or personal relationships that might influence the work reported herein. All authors gave their informed consent.

All authors have seen and approved the final version of the submitted manuscript. They guarantee that this article is the authors’ original work and has not been previously published or considered for publication elsewhere.

Additional information

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

Cite this article

Lu, H., Qian, L. & Shi, M. An infinite family of binary cyclic codes and two infinite families of ternary cyclic codes with good parameters.Cryptogr. Commun. 17, 1407–1425 (2025). https://doi.org/10.1007/s12095-025-00803-9

Download citation

Keywords

Mathematics Subject Classification (2010)