Abstract
Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ding et al. (IEEE Trans Inform Theory 57(12), 8000–8006, 2011); Ma et al. (IEEE Trans Inform Theory 57(1):397–402, 2011); Wang et al. (Trans Inf Theory 58(12):7253–7259, 2012); and Xiong (Finite Fields Appl 18(5):933–945, 2012). In this paper we use the method developed in Xiong (Finite Fields Appl 18(5):933–945, 2012) to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boston N., McGuire G.: The weight distributions of cyclic codes with two zeros and zeta functions. J. Symb. Comput. 45(7), 723–733 (2010).
Canteaut A., Charpin P., Dobbertin H.: Weight divisibility of cyclic codes, highly nonlinear functions on \(F_{2^m}\), and crosscorrelation of maximum-length sequences. SIAM J. Discret. Math. 13, 105–138 (2000).
Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15, 125–156 (1998).
Charpin P.: Cyclic codes with few weights and Niho exponents. J. Comb. Theory A 108(2), 247–259 (2004).
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).
Ireland K., Rosen M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990).
Koblitz N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97. Springer, New York (1984).
Luo J., Feng K.: On the weight distributions of two classes of cyclic codes. IEEE Trans. Inf. Theory 54(12), 5332–5344 (2008).
Luo J., Tang Y., Wang H.: On the weight distribution of a class of cyclic codes. In: ISIT 2009, Seoul, June 28–July 3 (2009).
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).
McGuire G.: On three weights in cyclic codes with two zeros. Finite Fields Appl. 10(1), 97–104 (2004).
Moisio M., Ranto K.: Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros. Finite Fields Appl. 13(4), 922–935 (2007).
Myerson G.: Period polynomials and Gauss sums for finite fields. Acta Arith. 39(3), 251–264 (1981).
Schoof R.: Families of curves and weight distribution of codes. Bull. Am. Math. Soc. 32(2), 171–183 (1995).
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).
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).
Acknowledgments
The author is grateful to the anonymous referees for many valuable suggestions which help improve the quality of the paper substantially, and to Professor Cunsheng Ding for bringing this problem to his attention. The author was supported by the Research Grants Council of Hong Kong under Project Nos. RGC606211 and DAG11SC02.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Q. Xiang.
Rights and permissions
About this article
Cite this article
Xiong, M. The weight distributions of a class of cyclic codes II. Des. Codes Cryptogr. 72, 511–528 (2014). https://doi.org/10.1007/s10623-012-9785-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-012-9785-0