Abstract
Recently, linear codes with a few weights have been extensively studied due to their applications in secret sharing schemes, authentication codes, constant composition codes. Results have shown that some optimal codes can be acquired if the defining sets are well chosen over finite fields. In this paper, we investigate the Lee-weight distribution of linear codes over the ring \(\mathbb {F}_p +u\mathbb {F}_p\) (p is an odd prime) based on defining sets by employing exponential sums. We then determine the explicit complete weight enumerator for the images of these linear codes under the Gray map. A class of constant weight codes that meets the Griesmer bound for constructing optimal constant composition codes achieving the LVFC bound is also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahn J., Ka D., Li C.: Complete weight enumerators of a class of linear codes. Des. Codes Cryptogr. 83(1), 1–17 (2017).
Anderson R., Ding C., Helleseth T., Kløve T.: How to build robust shared control systems. Des. Codes Cryptogr. 15(2), 111–124 (1998).
Andrews G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998).
Assmus E.F., Mattson H.F.: Coding and combinatorics. Siam Rev. 16(3), 349–388 (1974).
Bae S., Li C., Yue Q.: On the complete weight enumerators of some reducible cyclic codes. Discret. Math. 338(12), 2275–2287 (2015).
Byrne E., Greferath M., O’Sullivan M.E.: The linear programming bound for codes over finite Frobenius rings. Des. Codes Cryptogr. 42(3), 289–301 (2007).
Calderbank A.R., Goethals J.M.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984).
Carlet C., Ding C., Yuan J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005).
Ding C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015).
Ding K., Ding C.: Binary linear codes with three weights. IEEE Commun. Lett. 18(11), 1879–1882 (2014).
Ding K., Ding C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61(11), 5835–5842 (2015).
Ding C., Wang X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005).
Ding C., Helleseth T., Kløve T., X Wang: A general construction of authentication codes. IEEE Trans. Inf. Theory 53(6), 2229–2235 (2007).
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. Inf. Theory 40, 301–319 (1994).
Helleseth T., Kholosha A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006).
Kerdock A.M.: A class of low-rate nonlinear binary codes. Inf. Control 20(2), 182–187 (1972).
Li C., Yue Q., Li F.: Hamming weights of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 60(7), 3895–3902 (2014).
Li C., Yue Q., Fu F.W.: Complete weight enumerators of some cyclic codes. Des. Codes Cryptogr. 80(2), 295–315 (2016).
Li C., Bae S., Ahn J., Yang S., Yao Z.: Complete weight enumerators of some linear codes and their applications. Des. Codes Cryptogr. 81(1), 153–168 (2016).
Li F., Wang Q., Lin D.: A class of three-weight and five-weight linear codes. Discret. Appl. Math. (2017).
Li F., Wang Q., Lin D.: Complete weight enumerators of a class of three-weight linear codes. J. Appl. Math. Comput. 55(1–2), 733–747 (2017).
Lidl R., Niederreiter H., Cohn F.M.: Finite Fields. Cambridge University Press, Cambridge (1997).
Liu H., Maouche Y.: Several classes of trace codes with either optimal two weights or a few weights over $\mathbb{F}_{q}+u\mathbb{F}_{q}$. arXiv:1703.04968 [cs.IT].
Luo Y., Fu F., Vinck A.J.H., Chen W.: On constant-composition codes over $\mathbb{Z}_p$. IEEE Trans. Inf. Theory 49(11), 3010–3016 (2009).
Nechaev A.A.: Kerdock’s code in cyclic form. Diskr. Mat. 4, 123–139 (1989).
Preparata F.P.: A class of optimum nonlinear double-error-correcting codes. Inf. Control 13(4), 378–400 (1968).
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).
Shi M., Luo Y., Solé P.: Construction of one-Lee weight and two-Lee weight codes over $\mathbb{F}_p+ v\mathbb{F}_p$. J. Sys. Sci. Compl. 30(2), 484–493 (2017).
Sollé P.: Codes Over Rings. World Scientific Press, Singapore (2009).
Wang Q., Ding K., Xue R.: Binary linear codes with two weights. IEEE Commun. Lett. 19(7), 1097–1100 (2015).
Yang S., Yao Z.A.: Complete weight enumerators of a family of three-weight linear codes. Des. Codes Cryptogr. 82(3), 663–674 (2017).
Yang S., Yao Z.A., Zhao C.A.: A class of three-weight linear codes and their complete weight enumerators. Cryptogr. Commun. 9(1), 133–149 (2017).
Acknowledgements
The authors wish to thank the reviewers for valuable comments and suggestions which greatly helped us to improve this paper. This work is supported by the Natural Science Foundation of China with No. 11401408, Project of science and Technology Department of Sichuan Province with No. 2016JY0134.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.-L. Kim.
Rights and permissions
About this article
Cite this article
Liu, H., Liao, Q. Several classes of linear codes with a few weights from defining sets over \(\mathbb {F}_p+u\mathbb {F}_p\). Des. Codes Cryptogr. 87, 15–29 (2019). https://doi.org/10.1007/s10623-018-0478-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-018-0478-1