Abstract
In previous study, few-weight optimal codes were constructed by working on defining sets on the rings \({\mathbb {F}}_q[u]/(u^2)\), \({\mathbb {F}}_q[u]/(u^2-u)\) and \({\mathbb {F}}_q[u,v]/(u^2,v^2)\). We give a unified approach of this construction on a general \({\mathbb {F}}_q\)-algebra R. By applying this approach to the special case \(R={\mathbb {F}}_q[u]/(u^r)\), we construct many new classes of linear codes, decide their weight enumerators and show that they are new two or three-weight codes optimal or distance optimal to the Griesmer bound.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availibility
No dataset was generated or analyzed during this study.
References
Calderbank, A., Goethals, J.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984)
Ding, C., Wang, X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005)
Courteau, B., Wolfmann, J.: On triple sum sets and three-weight codes. Discrete Math. 50, 179–191 (1984)
Calderbank, A., Kantor, W.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986)
Wang, X., Zheng, D., Hu, L., Zeng, X.: The weight distributions of two classes of binary cyclic codes. Finite Fields Appl. 34, 192–207 (2015)
Mesnager, S., Sinak, A.: Several classes of minimal linear codes with few weights from weakly regular plateaued functions. IEEE Trans. Inf. Theory 66(4), 2296–2310 (2020)
Yuan, J., Carlet, C., Ding, C.: The weight distribution of a class of linear codes from perfect nonlinear functions. IEEE Trans. Inf. Theory 52(2), 712–717 (2006)
Vega, G.: A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field. Finite Fields Appl. 42, 23–38 (2016)
Wu, Y., Hyun, J.: Few-weight codes over \({{{\mathbb{F} }}_p}+u{{{\mathbb{F} }}_p}\) associated with down sets and their distance optimal Gray image. Discrete Appl. Math. 283, 315–322 (2020)
Hyun, J., Kim, H., Na, M.: Optimal non-projective linear codes constructed from down-sets. Discret. Appl. Math. 254, 135–145 (2019)
Shi, M., Li, X.: Two classes of optimal \(p\)-ary few-weight codes from down-sets. Discret. Appl. Math. 290, 60–67 (2021)
Shi, M., Liu, Y., Solé, P.: Optimal two-weight codes from trace codes over \({{\mathbb{F} }_2}+u{{\mathbb{F} }_2}\). IEEE Commun. Lett. 20(12), 2346–2349 (2016)
Liu, H., Maouche, Y.: Two or few-weight trace codes over \({{\mathbb{F} }_q}+u{{\mathbb{F} }_q}\). IEEE Trans. Inf. Theory 65(5), 2696–2703 (2019)
Shi, M., Li, X.: Few-weight codes over a non-chain ring associated with simplicial complexes and their distance optimal Gray image. Finite Fields Their Appl. 80, 101994 (2022)
Wu, Y., Zhu, X., Yue, Q.: Optimal few-weight codes from simplicial complexes. IEEE Trans. Inf. Theory 66(6), 3657–3663 (2020)
Shi, M., Wu, R., Liu, Y., Solé, P.: Two and three weight codes over \({{\mathbb{F} }_p}+u{{\mathbb{F} }_p}\). Cryptogr. Commun. 9(5), 637–646 (2017)
Shi, M., Guan, Y., Solé, P.: Two new families of two-weight codes. IEEE Trans. Inf. Theory 63(10), 6240–6246 (2017)
Luo, G., Cao, X.: Five classes of optimal two-weight linear codes. Cryptogr. Commun. 10(6), 1119–1135 (2018)
Shi, M., Liu, Y., Solé, P.: Optimal binary codes from trace codes over a non-chain ring. Discret. Appl. Math. 219, 176–181 (2017)
Liu, Y., Shi, M., Solé, P.: Two-weight and three-weight codes from trace codes over \({{\mathbb{F} }_p}+u{{\mathbb{F} }_p}+v{{\mathbb{F} }_p}+uv{{\mathbb{F} }_p}\). Discret. Math. 341(2), 350–357 (2018)
Yan, H., Zuo, K., Luo, G.: Infinite families of optimal linear codes and their applications to distributed storage systems. JAMC 1-17 (2022)
Griesmer, J.: A bound for error correcting codes. IBM J. Res. Dev. 4(6), 532–542 (1960)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge Univ. Press, Cambridge (1997)
Ma, C., Zeng, L., Liu, Y., Feng, D., Ding, C.: The weight enumerator of a class of cyclic codes. IEEE Trans. Inf. Theory. 57(1), 397–402 (2011)
Grassl, M.: Bounds on the minimum distance of linear codes. Available: http://www.codetables.de (2021)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Partially supported by Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302904) and Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200).
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
Wang, S., Ouyang, Y. & Xie, X. A unified construction of optimal and distance optimal two or three-weight codes. J. Appl. Math. Comput. 69, 2695–2715 (2023). https://doi.org/10.1007/s12190-023-01852-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-023-01852-0