Abstract
We construct new infinite families of (near) Plotkin-optimal 2-weight or 3-weight codes over \(\mathbb {Z}_4\) by using the shortening method. Furthermore, we completely determine the Lee weight distributions of our shortened codes. To achieve our goal, we use certain families of multivariable functions, and we interpret a shortening method followed by puncturing in terms of multivariable functions. According to this interpretation, we find explicit criteria for the shortened codes to have fewer Lee weights and larger minimum Lee weights after the shortening process. As our contribution, we emphasize that non-Plotkin-optimal code families are converted to Plotkin-optimal code families after the shortening process by using our main results. Furthermore, we produce new infinite families of (near) Plotkin-optimal 2-weight or 3-weight codes over \(\mathbb {Z}_4\), which extend the database of linear codes over \(\mathbb {Z}_4\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
No datasets were generated or analysed during the current study.
References
Aydin, N., Asamov, T., Yoshino, B.: A Database of Z4 codes [online], http://quantumcodes.info/Z4
Carlet, C.: One-weight \(\mathbb{Z}_{4}\)-linear codes. Coding theory, cryptography and related areas, 57–72, Springer, (2000)
Goethals, J.M.: Two dual families of nonlinear binary codes. Electron. Lett. 10, 471–472 (1974)
Hammons, A.R.Jr., 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)
Helgert, H.J., Stinaff, R.D.: Shortened BCH codes. IEEE Trans. Inform. Theory, IT-19, 818–820 (1973)
Huffman, W.C., Pless, V.: Fundamentals of Error Correcting codes. Cambridge University Press, (2003)
Hyun, J.Y., Han, N., Lee, Y.: The Griesmer codes of Belov type and optimal quaternary codes via multi-variable functions. Cryptogr. Commun. 16(3), 579–600 (2024)
Hyun, J.Y., Kim, B., Na, M.: Construction of minimal linear codes from multi-variable functions. Adv. Math. Commun. 15(2), 227–240 (2021)
Kerdock, A.M.: A class of low-rate nonlinear binary codes. Inf. Control 20, 182–187 (1972)
Kiermaier, M., Wassermann, A., Zwanzger, J.: New upper bounds on binary linear codes and a \(\mathbb{Z}_{4}\)-code with a better-than-linear gray image. IEEE Trans. Inform. Theory 62(12), 6768–6771 (2016)
Li, S., Shi, M.: Two infinite families of two-weight codes over \(\mathbb{Z}_{2^m}\). J. Appl. Math. Comput. 69(1), 201–218 (2023)
Liu, Y., Ding, C., Tang, C.: Shortened linear codes over finite fields. IEEE Trans. Inform. Theory 67(8), 5119–5132 (2021)
Preparata, F.P.: A class of optimum nonlinear double-error-correcting codes. Inf. Control 13, 378–400 (1968)
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. Inform. Theory 67(12), 7769–7781 (2021)
Shi, M., Kiermaier, M., Kurz, S., Solé, P.: Three-weight codes over rings and strongly walk regular graphs. Graphs Combin. 38(3), Paper No. 56, pp 23 (2022)
Shi, M., Liu, Y., Randriam, H., Sok, L.: P, Solé, Trace codes over \(\mathbb{Z}_4\), and Boolean functions. Des. Codes Cryptogr. 87(6), 1447–1455 (2019)
Shi, M., Solé, P.: Three-weight codes, triple sum sets, and strongly walk regular graphs. Des. Codes Cryptogr. 87(10), 2395–2404 (2019)
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)
Shi, M., Xuan, W., Solé, P.: Two families of two-weight codes over \(\mathbb{Z}_4\). Des. Codes Cryptogr. 88(12), 2493–2505 (2020)
Tang, H.C., Suprijanto, D.: A general family of Plotkin-optimal two-weight codes over \(\mathbb{Z}_{4}\). Des. Codes Cryptogr. 91(5), 1737–1750 (2023)
Tang, H.C., Suprijanto, D.: New optimal linear codes over \(\mathbb{Z}_4\). Bull. Aust. Math. Soc. 107(1), 158–169 (2023)
Wyner, A.D., Graham, R.L.: An upper bound on minimum distance for a \(k\)-ary code. Inf. Control 13, 46–52 (1968)
Zhu, X., Wu, Y., Yue, Q.: New quaternary codes derived from posets of the disjoint union of two chains. IEEE Comm. Lett. 24(1), 20–24 (2020)
Acknowledgements
The authors would like to thank the reviewers for their very helpful suggestions, which have improved the clarity of this paper. In particular, one of the reviewers mentioned important comments given in Remark 6.2.
Author information
Authors and Affiliations
Contributions
J.Y.H, J.J and Y.L. worked together on the concept and computation for main results, and all took part in writing the main text.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(NRF-2022R1A2C1003203).
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.
About this article
Cite this article
Hyun, J.Y., Jeong, J. & Lee, Y. Constructing optimal few weight quaternary linear codes via multivariable functions. Cryptogr. Commun. 17, 57–85 (2025). https://doi.org/10.1007/s12095-024-00739-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-024-00739-6