Abstract
Let \(p\) be an odd prime, and \(m\) and \(k\) be two positive integers with \(\frac{m}{\gcd (m,k)}\) being odd. This paper determines the weight distribution of a family of \(p\)-ary cyclic codes over \({\mathbb {F}}_p\) whose duals have three zeros \(\alpha ^{-2}, \alpha ^{-(p^{2k}+1)}\) and \(\alpha ^{-(p^{4k}+1)}\), where \(\alpha \) is a primitive element of \({\mathbb {F}}_{p^m}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ding C., Liu Y., Ma C., Zeng L.: The weight distributions of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 57(12), 8000–8006 (2011).
Ding C., Yang J.: Hamming weight in irrecducible codes. Discret. Math. 313(4), 434–446 (2013).
Feng K., Luo J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008).
Feng T.: On cyclic codes of length \(2^{2^r}-1\) with two zeros whose dual code have three weights. Des. Codes Cryptogr. 62(3), 253–258 (2012).
Kløve T.: Codes for Error Detection. World Scientific, Hackensack (2007).
Li S., Hu S., Feng T., Ge G.: The weight distribution of a class of cyclic codes related to Hermitian form graphs. IEEE Trans. Inf. Theory 59(5), 3064–3067 (2013).
Lidl R., Niederreiter H.: Finite Fields, Encyclopedia of Mathematics, vol. 20. Cambridge University Press, Cambridge (1983).
Luo J., Feng K.: Cyclic codes and sequences from generalized Coulter–Matthews function. IEEE Trans. Inf. Theory 54(12), 5345–5353 (2008).
Luo J., Feng K.: On the weight distributions of two classes of cyclic codes. IEEE Trans. Inf. Theory 54(12), 5332–5344 (2008).
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).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing, Amsterdam (1997).
McGuire G.: On three weights in cyclic codes with two zeros. Finite Fields Appl. 10(1), 97–104 (2004).
Rao A., Pinnawala N.: A family of two-weight irreducible cyclic codes. IEEE Trans. Inf. Theory 56(6), 2568–2570 (2010).
Vega G., Wolfmann J.: New classes of 2-weight cyclic codes. Des. Codes Cryptogr. 42(3), 327–334 (2007).
Vega G.: The weight distribution of an extended class of reducible cyclic codes. IEEE Trans. Inf. Theory 58(7), 4862–4869 (2012).
Wang B., Tang C., Qi Y., Yang Y., Xu M.: The weight distributions of cyclic codes and elliptic curves. IEEE Trans. Inf. Theory 58(12), 7253–7259 (2012).
Xiong M.: The weight distributions of a class of cyclic codes II. Des. Codes Crypt. (2012). doi:10.1007/s10623-012-9785-0.
Xiong M.: The weight distributions of a class of cyclic codes III. Finite Fields Appl. 21, 84–96 (2013).
Xiong M.: The weight distributions of a class of cyclic codes. Finite Fields Appl. 18(5), 933–945 (2012).
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).
Zeng X., Hu L., Jiang W., Yue Q., Cao X.: The weight distribution of a class of p-ary cyclic codes. Finite Fields Appl. 16(1), 56–73 (2010).
Zeng X., Shan J., Hu L.: A triple-error-correcting cyclic code from the Gold and Kasami–Welch APN power functions. Finite Fields Appl. 18(1), 70–92 (2012).
Zhou Z., Ding C.: A class of three-weight cyclic codes. Finite Fields Appl. 24, 79–93 (2013).
Zhou Z., Ding C., Luo J., Zhang A.: A family of five-weight cyclic codes and their weight enumerators. IEEE Trans. Inf. Theory 59(10), 6674–6682 (2013).
Acknowledgments
The authors wish to thank anonymous referees for their helpful comments, which have improved the presentation of this paper. The work was partially supported by National Natural Science Foundation of China under Grants 11101131, 61170257, 10990011 and the National Basic Research Program of China (2013CB834203).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Helleseth.
Rights and permissions
About this article
Cite this article
Zheng, D., Wang, X., Zeng, X. et al. The weight distribution of a family of \(p\)-ary cyclic codes. Des. Codes Cryptogr. 75, 263–275 (2015). https://doi.org/10.1007/s10623-013-9908-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-013-9908-2