Abstract
Linear codes with a few weights have nice applications in communication, secret sharing schemes, authentication codes, association schemes, block designs and so on. Projective binary linear codes are one of the most important subclasses of linear codes for practical applications. The objective of this paper is to construct projective binary linear codes with some special Boolean functions. Four families of binary linear codes with three or four weights are derived and the parameters of their duals are also determined. It turns out that the duals of these codes are optimal or almost optimal with respect to the sphere-packing bound. As applications, the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Anderson, R., Ding, C., Helleseth, T., Kløve, T.: How to build robust shared control systems. Des. Codes Cryptogr. 15, 111–124 (1998)
Ashikhmin, A., Barg, A.: Minimal vectors in linear codes. IEEE Trans. Inf. Theory 44, 2010–2017 (1998)
Blakley, G.R.: Safeguarding cryptographic keys. Proc. Natl. Comput. Conf. 48, 313–317 (1979)
Calderbank, R., Goethals, J.M.: Three-weight codes and assiociation schemes. Philips J. Res. 39, 143–152 (1984)
Calderbank, R., Kanter, W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986)
Coulter, R.S.: On the evaluation of a class of Weil sums in characteristic 2. N. Z. J. Math. 28, 171–184 (1999)
Delsarte, P.: Weights of linear codes and strongly regular normed spaces. Discrete Math. 3, 47–64 (1972)
Ding, C.: A construction of binary linear codes from Boolean functions. Discrete Math. 339, 2288–2303 (2016)
Ding, C.: Designs from Linear Codes. World Scientific, Singapore (2018)
Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 60, 3265–3275 (2015)
Ding, K., Ding, C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61, 5835–5842 (2015)
Ding, C., Li, C., Li, N., Zhou, Z.: Three-weight cyclic codes and their weight distributions. Discrete Math. 339, 415–427 (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.) Proceedings 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, 2274–2277 (2007)
Heng, Z., Ding, C., Zhou, Z.: Minimal linear codes over finite fields. Finite Fields Appl. 54, 176–196 (2018)
Heng, Z., Yue, Q.: Two classes of two-weight linear codes. Finite Fields Appl. 38, 72–92 (2016)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Lang, S.: Algebra. Addison-Wesley Publishing, Reading (1965)
Li, C., Bae, S., Ahn, J., Yang, S., Yao, Z.: Complete weight enumerators of some linear codes and their applications. Des. Codes Cryptogr. 81, 153–168 (2016)
Li, C., Yue, Q., Fu, F.: A construction of several classes of two-weight and three-weight linear codes. Appl. Algebra Eng. Commun. Comput. 28, 11–30 (2017)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)
Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, New York (2016)
Mesnager, S.: Linear codes with few weights from weakly regular bent functions based on a generic construction. Cryptogr. Commun. 9, 71–84 (2016)
Reed, I.S.: A class of multiple-error-correcting codes and the decoding scheme. IEEE Trans. Inf. Theory 4, 38–49 (1954)
Shamir, A.: How to share a secret. Commun. Assoc. Comput. Mach. 22, 612–613 (1979)
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, 1166–1176 (2016)
Tang, C., Qi, Y., Huang, M.: Two-weight and three-weight linear codes from square functions. IEEE Commun. Lett. 20, 29–32 (2016)
Xiang, C.: It is indeed a fundamental construction. arXiv:1610.06355 [cs.IT]
Yang, S., Zhao, C., Yao, Z.: A class of three-weight linear codes and their complete weight enumerators. Cryptogr. Commun. 9, 133–149 (2017)
Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52, 206–212 (2016)
Zhou, Z.: Three-weight ternary linear codes from a family of cyclic difference sets. Des. Codes Cryptogr. 86, 25133–2523 (2018)
Zhou, Z., Li, N., Fan, C., Helleseth, T.: Linear codes with two or three weights from quadratic Bent functions. Des. Codes Cryptogr. 81, 283–2952 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work of Z. Heng was supported in part by the National Science Foundation of China under Grant 11901049 and in part by the Fundamental Research Funds for the Central Universities, CHD, under Grant 300102129301. The work of Y. Wang was supported by the National Natural Science Foundation of China under Grant 61902304.
Rights and permissions
About this article
Cite this article
Heng, Z., Wang, W. & Wang, Y. Projective binary linear codes from special Boolean functions. AAECC 32, 521–552 (2021). https://doi.org/10.1007/s00200-019-00412-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-019-00412-z