Abstract
Let p be an odd prime, and m, k and d be positive integers such that \(2 \le k\le \frac{m+1}{2}\) and \(\hbox {gcd}(m,d)=1. \pi \) is a primitive element of the finite field \({\mathbb {F}}_{p^{m}}\). The weight enumerator of cyclic codes over \({\mathbb {F}}_{p}\) whose duals have 2k zeros \(\pi ^{-(p^{jd}+1)/2}\) and \(-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-1)\) is determined in the present paper. The weight enumerator of cyclic codes over \({\mathbb {F}}_{p}\) whose duals have \(2k-1\) zeros \(\pi ^{-(p^{(k-1)d}+1)/2}, \pi ^{-(p^{jd}+1)/2}\) and \(-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-2)\) is also determined when \(2\not \mid \frac{m}{gcd(m,k-1)}\) holds.
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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 Program 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.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008).
Feng T., Momihara K.: Evaluation of the weight distribution of a class of cyclic codes based on index 2 Gauss sums. IEEE Trans. Inf. Theory 59(9), 5980–5984 (2013).
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., Li N., Helleseth T., Ding C.: The weight distributions of several classes of cyclic codes from APN monomials. IEEE Trans. Inf. Theory 60(8), 4710–4721 (2014).
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. (2014). doi:10.1007/s10623-014-9984-y.
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).
Schmidt K.: Symmetric bilinear forms over finite fields with applications to coding theory, arXiv:1410.7184 (2014).
Sharma A., Bakshi G.: The weight distribution of some irreducible cyclic codes. Finite Fields Appl. 18(1), 144–159 (2012).
Trachtenberg H.M.: On the crosscorrelation functions of maximal linear recurring sequences. Ph.D. Dissertation, University of Southern California, Los Angels (1970).
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. Finite Fields Appl. 18(5), 933–945 (2012).
Yang J., Xiong M., Ding C., Luo J.: Weight distribution of a class of cyclic codes with arbitrary number of zeros. IEEE Trans. Inf. Theory 59(9), 5985–5993 (2013).
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).
Zheng D., Wang X., Zeng X., Hu L.: The weight distribution of a family of p-ary cyclic codes. Des. Codes Cryptogr. 75(2), 263–275 (2015).
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).
Zhu X., Yue Q., Hu L.: Weight distributions of cyclic codes of length \(l^m\). Finite Fields Appl. 31, 241–257 (2015).
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The authors are grateful to the referees for their careful reading of the original version of this paper, their detailed comments and suggestions, which have much improved the quality of this paper.
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Communicated by T. Helleseth.
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Yan, H., Liu, C. Two classes of cyclic codes and their weight enumerator. Des. Codes Cryptogr. 81, 1–9 (2016). https://doi.org/10.1007/s10623-015-0125-z
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DOI: https://doi.org/10.1007/s10623-015-0125-z