A note on the minimal binary linear code

Abstract

Due to the wide applications in communications, data storage and cryptography, linear codes have received much attention in the past decades. As a subclass of linear codes, minimal linear codes can be used to construct secret sharing with nice access structure. The objective of this paper is to construct new classes of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) from preferred binary linear codes, where \(w_{\min \limits }\) and \(w_{\max \limits }\) denote the minimum and maximum nonzero Hamming weights in \(\mathcal {C}\) respectively. Firstly, we introduce a concept called preferred binary linear codes and a class of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) can be deduced from preferred binary linear codes. As an application of preferred binary linear codes, we get a new class of six-weight minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }< 1/2\) from a known class of five-weight preferred binary linear codes. Secondly, by employing vectorial Boolean functions, we construct two new classes of preferred binary linear codes and, consequently, these two new classes of preferred binary linear codes can generate two new classes of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) and large minimum distance.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Anderson, R.J., Ding, C., Helleseth, T., Kløve, T.: How to build robust shared control systems. Des. Codes Cryptogr. 15(2), 111–124 (1998). https://doi.org/10.1023/A:1026421315292

    MathSciNet  Article  Google Scholar 

  2. 2.

    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). https://doi.org/10.1109/TIT.2015.2473861

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ding, C., Wang, X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005). https://doi.org/10.1016/j.tcs.2004.09.011

    MathSciNet  Article  Google Scholar 

  4. 4.

    Delsarte, P., Levenshtein, V. I.: Association schemes and coding theory. IEEE Trans. Inf. Theory 44(6), 2477–2504 (1998). https://doi.org/10.1109/18.720545

    MathSciNet  Article  Google Scholar 

  5. 5.

    Ashikhmin, A., Barg, A.: Minimal vectors in linear codes. IEEE Trans. Inf. Theory 44(5), 2010–2017 (1998)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015). https://doi.org/10.1109/TIT.2015.2420118

    MathSciNet  Article  Google Scholar 

  7. 7.

    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)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Tang, C., Qi, Y., Huang, D.: Two-weight and three-weight linear codes from square functions. IEEE Commun. Lett. 20(1), 29–32 (2016). https://doi.org/10.1109/LCOMM.2015.2497344

    Article  Google Scholar 

  9. 9.

    Heng, Z., Yue, Q.: A class of binary linear codes with at most three weights. IEEE Commun. Lett. 19(9), 1488–1491 (2015). https://doi.org/10.1109/LCOMM.2015.2455032

    Article  Google Scholar 

  10. 10.

    Heng, Z., Yue, Q., Li, C.: Three classes of linear codes with two or three weights. Discret. Math. 339(11), 2832–2847 (2016). https://doi.org/10.1016/j.disc.2016.05.033

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chang, S., Hyun, J. Y.: Linear codes from simplicial complexes. Linear codes from simplicial, pp. 1–15 (2017)

  12. 12.

    Ding, C., Heng, Z., Zhou, Z.: Minimal binary linear codes. IEEE Transactions on Information Theory (2018)

  13. 13.

    Zhang, W., Yan, H., Wei, H.: Four families of minimal binary linear codes with wmin/wmax ≤ 1/2. Appl. Algebra Eng. Commun. Comput. 30(2), 175–184 (2019)

    Article  Google Scholar 

  14. 14.

    Bartoli, D., Bonini, M.: Minimal linear codes in odd characteristic. IEEE Trans. Inf. Theory 65(7), 4152–4155 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Bonini, M., Borello, M.: Minimal linear codes arising from blocking sets. arXiv:1907.04626 (2019)

  16. 16.

    Heng, Z., Ding, C., Zhou, Z.: Minimal linear codes over finite fields. Finite Fields Appl. 54, 176–196 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Ding, C.: A construction of binary linear codes from Boolean functions. Discret. Math. 339(9), 2288–2303 (2016). https://doi.org/10.1016/j.disc.2016.03.029

    MathSciNet  Article  Google Scholar 

  18. 18.

    Li, C., Yue, Q., Fu, F. W.: A construction of several classes of two-weight and three-weight linear codes. Applicable Algebra in Engineering, Communication and Computing, pp. 1–20 (2016)

  19. 19.

    Mesnager, S.: Linear codes with few weights from weakly regular bent functions based on a generic construction. Cryptography and Communications, pp. 1–14. https://doi.org/10.1007/s12095-016-0186-5 (2015)

  20. 20.

    Xu, G., Cao, X., Xu, S.: Two classes of p-ary bent functions and linear codes with three or four weights. Cryptogr. Commun. 9(1), 117–131 (2017)

    MathSciNet  Article  Google Scholar 

  21. 21.

    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). https://doi.org/10.1007/s10623-015-0144-9

    MathSciNet  Article  Google Scholar 

  22. 22.

    Nyberg, K.: Perfect Nonlinear S-Boxes. In: Advances in Cryptology<a̱EUROCRYPT< 91, pp. 378–386. Springer (1991)

  23. 23.

    Carlet, C., Ding, C.: Nonlinearities of S-boxes. Finite Fields Appl. 13(1), 121–135 (2007)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Wadayama, T., Hada, T., Wakasugi, K., Kasahara, M.: Upper and lower bounds on maximum nonlinearity of n-input m-output Boolean function. Des. Codes Crypt. 23(1), 23–34 (2001)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Tang, D., Carlet, C., Zhou, Z.: Binary linear codes from vectorial Boolean functions and their weight distribution. Discret. Math. 340(12), 3055–3072 (2017)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous reviewers for their valuable suggestions and comments, which improved the quality of our paper. The work of Deng Tang was supported by the National Natural Science Foundation of China (grants 61872435 and 61602394).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xia Li.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the Topical Collection on Special Issue on Sequences and Their Applications

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tang, D., Li, X. A note on the minimal binary linear code. Cryptogr. Commun. 12, 375–388 (2020). https://doi.org/10.1007/s12095-019-00412-3

Download citation

Keywords

  • Linear codes
  • Minimal codes
  • Boolean functions
  • Preferred binary linear codes

Mathematics Subject Classification (2010)

  • 06E30
  • 11T71
  • 94A60