Abstract
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach. The first objective of this paper is to establish relationships among some classes of linear codes obtained with this approach, so that the parameters of some classes of linear codes can be derived from those of other classes with known parameters. In this way, linear codes with new parameters will be derived. The second is to present a class of three-weight binary codes and consider their applications in secret sharing.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, R., Ding, C., Helleseth, T., Kløve, T.: How to build robust shared control systems. Des. Codes Cryptogr. 15(2), 111–124 (1998)
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)
Carlitz, L.: Explicit evaluation of certain exponential sums. Mathematica Scandinavics 44, 5–16 (1979)
Coulter, R.S.: On the evaluation of a class of Weil sums in characteristic 2. New Zerland J. of Math. 28, 171–184 (1999)
Dickson, L.E.: Linear Groups with an Exposition of the Galois Field Theory. Dover, New York (1958)
Ding, C.: A class of three-weight and four-weight codes. In: Xing, C., et al. (eds.) Proc. of the Second International Workshop on Coding Theory and Cryptography, Lecture Notes in Computer Science, vol. 5557, pp 34–42. Springer, Berlin (2009)
Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 60(6), 3265–3275 (2015)
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, 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, to appear in IEEE Trans. Inf. Theory. arXiv: 1503.06512
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)
Wang, Q., Ding, K., Xue, R.: Binary linear codes with two weights. IEEE Commun. Lett. 19, 1097–1100 (2015)
Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52(1), 206–212 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xiang, C. Linear codes from a generic construction. Cryptogr. Commun. 8, 525–539 (2016). https://doi.org/10.1007/s12095-015-0158-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-015-0158-1