Abstract
For positive integers k ≥ 2 and t, let m = 2kt and α be a primitive element of the finite field \(\mathbb {F}_{2^{m}}\). In this paper, we study the parameters of a class of cyclic codes \(\mathcal {C}_{(1,v)}\) which has two zeros α and αv with \(v=\frac {2^{m}-1}{2^{t}+1}\). It is shown that \(\mathcal {C}_{(1,v)}\) is optimal or almost optimal with respect to the sphere packing bound. Based on some results of Kloosterman sums and Gaussian periods, the weight distribution of the dual code of \(\mathcal {C}_{(1,v)}\) is completely determined when t = 5.
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References
Cao, X.: A note on the moments of Kloosterman sums. Appl. Algebra Engrg. Comm. Comput. 20(5-6), 447–457 (2009)
Cao, X., Hu, L.: A construction of hyperbent functions with polynomial trace form. Sci. China Math. 54(10), 2229–2234 (2011)
Carlet, C., Ding, C., Yuan, J.: Linear codes from highly nonlinear functions and their secret sharing schemes. IEEE Trans. Inform. Theory 51(6), 2089–2102 (2005)
Delsarte, P.: On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inform. Theory 21(5), 575–576 (1975)
Dinh, H. Q., Wang, X., Liu, H., Sriboonchitta, S.: On the Hamming distances of repeated- root consta cyclic codes of length 4ps. Discret. Math. 342(5), 1456–1470 (2019)
Ding, C., Li, C., Li, N., Zhou, Z.: Three-weight cyclic codes and their weight distributions. Discret. Math. 339(2), 415–427 (2016)
Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inform. Theory 59(9), 5898–5904 (2013)
Ding, C., Wang, X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005)
Ding, C., Yang, Y., Tang, X.: Optimal sets of frequency hopping sequences from linear cyclic codes. IEEE Trans. Inform. Theory 56(7), 3605–3612 (2010)
Ding, C., Yang, J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313(4), 434–446 (2013)
Dobbertin, H., Helleseth, T., Kumar, V., Martinsen, H.: Ternary m-sequences with three-valued cross-correlation function: New decimations of Welch and Niho type. IEEE Trans. Inform. Theory 47(4), 1473–1481 (2001)
Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans Inform. Theory 52(5), 2018–2032 (2006)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Kløve, T.: Codes for Error Detection, World Scientific. World Scientific Publishing Co., Inc., New Jersey (2007)
Lachaud, G., Wolfmann, J.: The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inform. Theory 36(3), 686–692 (1990)
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)
Lint, J.H.: Introduction to Coding Theory. Springer, Berlin (1999)
Liu, K., Ren, W., Wang, F., Wang, J.: A new class of distance-optimal binary cyclic codes and their duals. Appl. Algebra Engin. Commun. Comput. https://doi.org/10.1007/s00200-020-00478-0 (2021)
Luo, R., Wei, L., Cheng, F., Du, X.: A class of binary cyclic codes with four weights. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 100-A, 965–968 (2017)
Shi, M., Li, X., Neri, A., Solė, P.: : How many weights can a cyclic code have? IEEE Trans. Inform. Theory 66(3), 1449–1459 (2020)
Yang, J., Xiong, M., Ding, C., Luo, J.: Weight distribution of a class of cyclic codes with arbitrary number of zeros. IEEE Trans. Inform. Theory 59(9), 5985–5993 (2013)
Zeng, X., Fan, C., Zeng, Q., Qi, Y.: Two classes of binary cyclic codes and their weight distributions. Appli. Algebra Engin., Commun. Comput. 32, 49–61 (2021)
Acknowledgments
The authors are grateful to the two anonymous reviewers for careful reading and for many valuable comments that improved the quality of the paper. Q. Wang was partially supported by the National Natural Science Foundation of China under Grant no. 11931005 and the Shenzhen fundamental research programs under Grant no.20200925154814002. H. Yan was partially supported by the National Natural Science Foundation of China under Grant no.11801468 and the Fundamental Research Funds for the Central Universities of China under Grant no.2682021ZTPY076.
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Liu, K., Wang, Q. & Yan, H. A class of binary cyclic codes with optimal parameters. Cryptogr. Commun. 14, 663–675 (2022). https://doi.org/10.1007/s12095-021-00548-1
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DOI: https://doi.org/10.1007/s12095-021-00548-1