Binary linear codes with two or three weights from niho exponents
Tóm tắt
Linear codes with few weights have applications in secret sharing, authentication codes, association schemes, date storage systems, strongly regular graphs and some other fields. In this paper, we present several classes of binary linear codes with two or three weights and study their weight distributions. Two classes of strongly regular graphs are constructed from binary linear codes with two weights. Numerical results show that some of the obtained codes are either optimal or near optimal with respect to certain bounds on linear codes.
Tài liệu tham khảo
Anderson, R., Ding, C., Helleseth, T., Klove, T.: How to build robust shared control systems. Des. Codes Crypt. 15(2), 111–124 (1998)
Calderbank, A., Goethals, J.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984)
Calderbank, A., Kantor, W.: The geometry of two-weight codes. Bull. London Math. Soc. 18, 97–122 (1986)
Carlet, C., Ding, C., Yuan, J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005)
Coulter, R.: On the evaluation of a class of Weil sums in characteristic 2. N. Z. J. Math. 28, 171–184 (1999)
Ding, C., Luo, J., Niederreiter, H.: Two-weight codes punctured from irreducible cyclic codes. Ser. Coding Theory Cryptol. 4, 119–124 (2008)
Ding, C., Niederreiter, H.: Cyclotomic linear codes of order 3. IEEE Trans. Inf. Theory 53 (6), 2274–2277 (2007)
Ding, C., Wang, X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 30(1), 81–99 (2005)
Ding, C.: The weight distribution of some irreducible cyclic codes. IEEE Trans. Inf. Theory 55 (3), 955–960 (2009)
Ding, C.: A construction of binary linear codes from Boolean functions. Discret. Math. 339, 2288–2303 (2016)
Ding, C., Helleseth, T.: New generalized cyclotomy and its applications. Finite Fields Appl. 4(2), 140–166 (1998)
Dobbertin, H., Felke, P., Helleseth, T., Rosendahl, P.: Niho type cross-correlation functions via Dickson polynomials and Kloosterman sums. IEEE Trans. Inf. Theory 52(2), 613–627 (2006)
Ding, K., Ding, C.: Binary linear codes with three weights. IEEE Commun. Lett. 18(11), 1879–1882 (2014)
Ding, K., Ding, C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61(11), 5835–5842 (2015)
Huffman, W., Pless, V.: Fundamentals of error-correcting codes. Cambridge university press (2003)
Li, C., Yue, Q., Li, F.: Hamming weights of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 60(7), 3895–3902 (2014)
Li, C., Yue, Q., Fu, F.: Complete weight enumerators of some cyclic codes. Des. Codes Crypt. 338(12), 1–21 (2015)
Lidl, R., Niederreiter, H.: Finite fields. Cambridge university press (1997)
Ma, S.: A survey of partial difference sets. Des. Codes Crypt. 4(3), 221–261 (1994)
Niho, Y.: Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences. University of Southern Califorlia, Los Angeles (1972). Phd Thesis
Qi, Y., Tang, C., Huang, D.: Binary linear codes with few weights. IEEE Commun. Lett. 20(2), 208–211 (2016)
Tang, C., Li, N., Qi, Y., Zhou, Z.: Linear codes with two or three weights from weakly regular bent functions. IEEE Trans. Inf. Theory 62(3), 1166–1176 (2015)
Xiang, C.: It is indeed a fundamental construction of all linear codes. arXiv:1610.06355 (2016)
Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52(1), 206–212 (2006)
Yang, S., Yao, Z.: Complete weight enumerators of a family of three-weight linear codes. Des. Codes Crypt. doi:10.1007/s10623-016-0191-x (2015)
Yang, S., Yao, Z., Zhao, C.: A class of three-weight linear codes and their complete weight enumerators. Cryptogr. Commun. doi:10.1007/s12095-016-0187-4 (2016)
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)
Zhou, Z., Li, N., Fan, C., Helleseth, T.: Linear codes with two or three weights from quadratic Bent functions. Des. Codes Crypt. 81(2), 283–295 (2016)
Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Appl. 25(10), 79–93 (2013)