Skip to main content
SpringerLink
Log in

A general family of Plotkin-optimal two-weight codes over \(\mathbb {Z}_4\)

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

Abstract

We obtained all possible parameters of Plotkin-optimal two-Lee weight projective codes over \(\mathbb {Z}_4,\) together with their weight distributions. We show the existence of codes with these parameters as well as their weight distributions by constructing an infinite family of two-weight codes. Previously known codes constructed by Shi et al. (Des Codes Cryptogr 88(12): 2493–2505, 2020) can be derived as a special case of our results. We also prove that the Gray image of any Plotkin-optimal two-Lee weight projective codes over \(\mathbb {Z}_4\) has the same parameters and weight distribution as some two-weight binary projective codes of type SU1 in the sense of Calderbank and Kantor (Bull Lond Math Soc 18: 97–122, 1986).

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. Calderbank R.: On uniformly packed \([n, n-k,4]\) codes over \(GF(q)\) and a class of caps in \(PG(k-1, q),\). J. Lond. Math. Soc. 26(2), 365–384 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  2. Calderbank R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlet C.: One-weight \({\mathbb{Z} }\)-linear codes. In: Buchmann J., et al. (eds.) Coding Theory, Cryptography and Related Areas. Springer, New York (2000).

    Google Scholar 

  4. Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret Math. 3, 47–62 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  5. Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \({\mathbb{Z} }_4\)-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory 40(2), 301–319 (1994).

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  7. Kløve T.: Support weight distribution of linear codes. Discret Math. 106–107, 311–316 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  8. Li S., Shi M.: Two infinite families of two-weight codes over \({\mathbb{Z} }_{2^m}\). J. Appl. Math. Comput. (2022). https://doi.org/10.1007/s12190-022-01736-9.

    Article  Google Scholar 

  9. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977).

    MATH  Google Scholar 

  10. Shi M., Guan Y., Solé P.: Two new families of two-weight codes. IEEE Trans. Inf. Theory 63(10), 6240–6246 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  11. Shi M., Honold T., Solé P., Qiu Y., Wu R., Sepasdar Z.: The geometry of two-weight codes over \({\mathbb{Z} }_{p^m}\). IEEE Trans. Inf. Theory 67(12), 7769–7781 (2021).

    Article  MATH  Google Scholar 

  12. Shi M., Sepasdar Z., Alahmadi A., Solé P.: On two-weight \({\mathbb{Z} }_{2^k}\)-codes. Des. Codes Cryptogr. 86(6), 1201–1209 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  13. Shi M., Wang Y.: Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over \({\mathbb{Z} }_4,\). J. Syst. Sci. Complex. 27(4), 795–810 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  14. Shi M., Xu L.L., Yang G.: A note on one weight and two weight projective \({\mathbb{Z} }_4\)-codes. IEEE Trans. Inf. Theory 63(1), 177–182 (2017).

    Article  Google Scholar 

  15. Shi M., Xuan W., Solé P.: Two families of two-weight codes over \({\mathbb{Z} }_4\). Des. Codes Cryptogr. 88(12), 2493–2505 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  16. Tang H.C., Suprijanto D.: New optimal linear codes over \({\mathbb{Z} }_4\). Bull. Aust. Math. Soc. (2022). https://doi.org/10.1017/S0004972722000399.

    Article  Google Scholar 

  17. Wan Z.: Quaternary Codes. World Scientifc, Singapore (1997).

    Book  MATH  Google Scholar 

  18. Wyner A.D., Graham R.L.: An upper bound on minimum distance for a \(k\)-ary code. Inf. Control 13(1), 46–52 (1968).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank two anonymous referees for their meticulous reading of the manuscript. Their suggestions have been very valuable in improving the presentation of this paper. This research is supported by Institut Teknologi Bandung (ITB) and the Ministry of Education, Culture, Research, and Technology (Kementerian Pendidikan, Kebudayaan, Riset, dan Teknologi (Kemdikbudristek)), Republic of Indonesia.

Author information

Authors and Affiliations

  1. Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, Indonesia

    Hopein Christofen Tang & Djoko Suprijanto

Authors
  1. Hopein Christofen Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Djoko Suprijanto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Djoko Suprijanto.

Additional information

Communicated by J.-L. Kim.

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

Tang, H.C., Suprijanto, D. A general family of Plotkin-optimal two-weight codes over \(\mathbb {Z}_4\). Des. Codes Cryptogr. 91, 1737–1750 (2023). https://doi.org/10.1007/s10623-022-01176-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-022-01176-3

Keywords

Mathematics Subject Classification

Search

Navigation