Abstract
Linear codes have been an interesting subject of study for many years, as linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a class of linear codes with a few weights over the finite field GF(p) are presented and their weight distributions are also determined, where p is an odd prime. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes.
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Anderson, R., Ding, C., Helleseth, T., Kløve, T.: How to build robust shared control systems. Des. Codes Cryptogr. 15(2), 111–124 (1998)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comput. 24(3), 235–265 (1997)
Calderbank, A.R., Goethals, J.M.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984)
Calderbank, A.R., Kantor, W.M.: 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.S.: Further evaluations of Weil sums. Acta Arith 86(3), 217–226 (1998)
Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 60(6), 3265–3275 (2015)
Ding, C., et al.: A class of three-weight and four-weight codes. In: Xing, C. (ed.) Proc. of the Second International Workshop on Coding Theory and Cryptography, Lecture Notes in Computer Science, vol. 5557, pp. 34–42. Springer Verlag, Berlin (2009)
Ding, C.: A construction of binary linear codes from boolean functions. Discret. Math. 339(9), 2288–2303 (2016)
Ding, C., Luo, J., Niederreiter, H.: Two weight codes punctured from irreducible cyclic codes. In: Li, Y., Ling, S., Niederreiter, H., Wang, H., Xing, C., Zhang, S. (eds.) Proc. of the First International Workshop on Coding Theory and Cryptography, pp. 119–124. World Scientific, Singapore (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. 330, 81–99 (2005)
Ding, C., Li, C., Li, N., Zhou, Z.: Three-weight cyclic codes and their weight distributions. Discret. Math. 339(2), 415–427 (2014)
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)
Feng, K., Luo, J.: Value distribution of exponential sums from perfect nonlinear functions and their applications. IEEE Trans. Inform. Theory 53(9), 3035–3041 (2007)
Huffman, W. C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd ed, vol. 84. Springer-Verlag, New York (1990). Graduate Texts in Mathematics
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)
Li, F., Wang, Q., Lin, D.: A class of three-weight and five-weight linear codes. arXiv:1509.06242v1 (2015)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)
Mesnager, S.: Linear codes with few weights from weakly regular bent functions based on a generic construction. IACR Cryptology ePrint Archive, 1103 (2015)
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., Helleseth, T.: Linear codes with two or three weights from weakly regular bent functions. IEEE Trans. Inf. Theory 62(3), 1166–1176 (2016)
Tang, C., Qi, Y., Huang, D.: Two-weight and three-weight linear codes from square functions. IEEE Commun. Lett. 20(1), 29–32 (2016)
Wang, Q., Ding, K., Xue, R.: Binary linear codes with two weights. IEEE Commun. Lett. 19(7), 1097–1100 (2015)
Xiang, C.: Linear codes from a generic construction. Cryptogr. Commun. 8(4), 525–539 (2016)
Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52(1), 206–212 (2006)
Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014)
Zhou, Z., Li, N., Fan, C., Helleseth, T.: Linear codes with two or three weights from quadratic bent functions, Des. Codes Cryptogr. doi:10.1007/s10623-015-0144-9 (2015)
Acknowledgments
The authors are very grateful to the reviewers and the Editor, for their comments and suggestions that improved the presentation and quality of this paper. The research of K. Feng was supported by NSFC No. 11471178, 11571007 and the Tsinghua National Lab. for Information Science and Technology. The research of C. Tang was supported by NSFC No. 11401480, 11531002. C. Tang also acknowledges support from 14E013 and CXTD2014-4 of China West Normal University.
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Xiang, C., Tang, C. & Feng, K. A class of linear codes with a few weights. Cryptogr. Commun. 9, 93–116 (2017). https://doi.org/10.1007/s12095-016-0200-y
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DOI: https://doi.org/10.1007/s12095-016-0200-y