Abstract
In the past decades, linear codes with a few weights have been extensively studied for their applications in space communication, data storage and cryptography etc. We construct several classes of binary linear codes and determine their weight distributions. Most of these codes can be used in secret sharing schemes.
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Blake, I.F., Kith, K.: On the complete weight enumerator of Reed–Solomon codes. SIAM J. Disc. Math. 4(2), 164–171 (1991)
Carlitz, L.: Exolicite evaluation of certain exponential sums. Math. Scand. 44, 5–16 (1979)
Carlitz, L.: Evaluation of some exponential sums over a finite field. Math. Nachr. 96, 319–339 (1980)
Coulter, R.S.: Explicit evaluation of some Weil sums. Acta Arith. 83, 241–251 (1998)
Coulter, R.S.: On the evaluation of a class of Weil sums in characteristic 2. N. Z. J. Math. 28, 171–184 (1999)
Choi, S.T., Kim, J.Y., No, J.S., Chung, H.: Weight distribution of some cyclic codes. In: Proceedings of the International Symposium on Information Theory, pp. 2911–2913 (2012)
Courteau, B., Wolfmann, J.: On triple-sum-sets and two or three weights codes. Discrete Math. 50, 179–191 (1984)
Ding, C.: A class of three-weight and four-weight codes. In: Xing, C., et al. (eds.) Proceedings of the Second International Workshop on Coding Theory and Cryptography, Lecture Notes in Computer Science. Springer, vol. 5557, pp. 34–42(2009)
Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015)
Ding, K., Ding, C.: Bianry 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)
Ding, C., Gao, Y., Zhou, Z.: Five families of three-weight ternary cyclic codes and their duals. IEEE Trans. Inf. Theory 59(12), 7940–7946 (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)
Ding, C., Luo, J., Niederreiter, H.: Two-weight codes punctured from irreducible cyclic codes. In: Li, Y., et al. (eds.) Proceedings of the First Worshop on Coding and Cryptography, pp. 119–124. World Scientific, Singapore (2008)
Ding, C., Li, C., Li, N., Zhou, Z.: Three-weight cyclic codes and their weight distributions. Disctrete Math. 339(2), 415–427 (2016)
Ding, C., Yang, J.: Hamming weights in irreducible cyclic codes. Discrete Math. 313(4), 434–446 (2013)
Ding, C., Niederreiter, H.: Cyclotomic linear codes of order \(3\). IEEE Trans. Inf. Theory 53(6), 2274–2277 (2007)
Du, X., Wan, Y.: Linear codes from quadratic forms. Appl. Algebra Eng. Commun. Comput. 28(6), 535–547 (2017)
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., Leung, K., Xiang, Q.: Binary cyclic codes with two primitive nonzeros. Sci. China Math. 56(7), 1403–1412 (2012)
Hou, X.: Explicit evaluation of certain exponential sums of binary quadratic functions. Finite Fields Appl. 13, 843–868 (2007)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Kith, K.: Complete weight enumeration of Reed-Solomon codes, Masters thesis, Department of Electrical and Computing Engineering, University of Waterloo, Waterloo, Ontario, Canada (1989)
Luo, J., Feng, K.: On the weight distribution of two classes of cyclic codes. IEEE Trans. Inf. Theory 54(12), 5332–5344 (2008)
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)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, New York (1997)
Qi, Y., Tang, C., Huang, D.: Binary linear codes with few weights. IEEE Commun. Lett. 20(2), 208–211 (2016)
Tang, C., Xiang, C., Feng, K.: Linear codes with few weights from inhomogeneous quadratic functions. Des. Codes Cryptogr. 83(3), 691–714 (2017)
Wang, Q., Ding, K., Xue, R.: Binary linear codes with two weights. IEEE Commun. Lett. 19(7), 1097–1100 (2015)
Wang, Q., Ding, K., Lin, D., Xue, R.: A kind of three-weight linear codes. Cryptogr. Commun. 9(3), 315–322 (2017)
Yang, S., Kong, X., Tang, C.: A construction of linear codes and their complete weight enumerators. Finite Fields Their Appl. 48, 196–226 (2017)
Yang, S., Yao, Z.-A., Zhao, C.-A.: A class of three-weight linear codes and their complete weight enumerators. Cryptogr. Commun. 9, 133–149 (2017)
Yang, S., Yao, Z.-A.: Complete weight enumerators of a family of three-weight linear codes. Des. Codes Cryptogr. 82(3), 663–674 (2017)
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)
Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52(1), 206–212 (2006)
Zhang, D., Fan, C., Peng, D., Tang, X.: Complete weight enumerators of some linear codes from quadratic forms. Cryptogr. Commun. 9, 151–163 (2017)
Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Their 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. 81(2), 283–295 (2016)
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This research is supported by National Natural Science Foundation of China (61602342).
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Li, F., Yan, Y., Wang, Q. et al. Several classes of binary linear codes and their weight enumerators. AAECC 30, 93–106 (2019). https://doi.org/10.1007/s00200-018-0361-3
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DOI: https://doi.org/10.1007/s00200-018-0361-3