Abstract
In this paper, a family of six-weight cyclic codes over \(\mathbb {F}_{p}\) whose duals have three zeros is presented, where \(p\) is an odd prime. Furthermore, the weight distributions of these cyclic codes are determined.
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References
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 Prog. Rep. 16, 128–131 (1973).
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 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.: Value distributions of exponential sums from perfect nonlinear functions and their applications. IEEE Trans. Inf. Theory 53(7), 3035–3041 (2007).
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).
Lidl R., Niederreiter H.: Finite Fields. Addison-Wesley, Boston (1983).
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).
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).
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).
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).
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The authors are very grateful to the Editor in Chief, the Associate Editor and the reviewers, for their comments that improved the quality of this paper. This work is supported by the National Natural Science Foundation of China (No. 11071160).
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Communicated by C. Carlet.
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Liu, Y., Yan, H. & Liu, C. A class of six-weight cyclic codes and their weight distribution. Des. Codes Cryptogr. 77, 1–9 (2015). https://doi.org/10.1007/s10623-014-9984-y
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DOI: https://doi.org/10.1007/s10623-014-9984-y