Abstract
In this paper, a family of cyclic codes over \({\mathbb {F}}_{p}\) whose duals have five zeros is presented, where p is an odd prime. Furthermore, the weight distribution of these cyclic codes is determined.
Similar content being viewed by others
References
Antweiler M., Bomer L.: Complex sequences over \(GF(p^{M})\) with a two-level autocorrelation function and a large linear span. IEEE Trans. Inf. Theory 38, 120C130 (1992).
Baumert L.D., McEliece R.J.: Weights of irreducible cyclic codes. Inf. Control 20(2), 158–175 (1972).
Baumert L.D., Mykkeltveit J.: Weight distribution of some irreducible cyclic codes. DSN Progr. Rep. 16, 128–131 (1973).
Carlet C., Ding C., Yuan J.: Linear codes from highly nonlinear functions and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005).
Ding C.: The weight distribution of some irreducible cyclic codes. IEEE Trans. Inf. Theory 55(3), 955–960 (2009).
Ding C., Yang J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313(4), 434–446 (2013).
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).
Feng T.: On cyclic codes of length \(2^{2^{r}}-1\) with two zeros whose dual codes have three weights. Des. Codes Cryptogr. 62, 253–258 (2012).
Feng K., Luo J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008).
Feng T., Leung K., Xiang Q.: Binary cyclic codes with two primitive nonzeros. Sci. China Math. 56(7), 1403–1412 (2012).
Li C., Yue Q.: Weight distributions of two classes of cyclic codes with respect to two distinct order elements. IEEE Trans. Inf. Theory 60(1), 296–303 (2014).
Li C., Yue Q., Li F.: Weight distributions of cyclic codes with respect to pairwise coprime order elements. Finite Fields Appl. 28, 94–114 (2014).
Li F., Yue Q., Li C.: The minimum Hamming distances of irreducible cyclic codes. Finite Fields Appl. 29, 225–242 (2014).
Lidl R., Niederreiter H.: Finite Fieds. Addison-Wdsley, Reading (1983).
Liu Y., Yan H.: A class of five-weight cyclic codes and their weight distribution. Des. Codes Cryptogr. (2015). doi:10.1007/s10623-015-0056-8.
Liu Y., Yan H., Liu C.: A class of six-weight cyclic codes and their weight distribution. Des. Codes Cryptogr. 77(1), 1–9 (2015).
Luo J., Feng K.: Cyclic codes and sequences form generalized Coulter–Matthews function. IEEE Trans. Inf. Theory 54(12), 5345–5353 (2008).
Luo J., Feng K.: On the weight distribution 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).
Trachtenberg H.M.: On the crosscorrelation functions of maximal linear recurring sequences. Ph.D. dissertation, University of South California, Los Angeles (1970).
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).
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).
Zhou Z., Ding C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014).
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 are very grateful to the Editor in Chief and the anonymous reviewers for their valuable comments and suggestions that have improved the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Ding.
Rights and permissions
About this article
Cite this article
Liu, Y., Liu, C. A class of cyclic codes whose duals have five zeros. Des. Codes Cryptogr. 81, 225–238 (2016). https://doi.org/10.1007/s10623-015-0138-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-015-0138-7