Minimal linear codes with six-weights based on weakly regular plateaued balanced functions

Abstract

Constructing minimal linear codes has a great interest in coding theory since they have an important role in describing access structures in secret sharing schemes and they are employed to design secure two-party computation protocols. Many methods of constructing linear codes have been proposed in the literature, and the most famous one is based on functions over finite fields. Linear codes derived from cryptographic functions have desirable algebraic structures that are significant from the application point of view. We in this paper study the construction of linear codes from some cryptographic functions over finite fields. We aim to construct new minimal codes by using a new type of function in the known construction method. To do this, we propose to use new subsets of the pre-images of weakly regular plateaued balanced functions. We then obtain five infinite classes of six-weight minimal codes from five different subsets of the pre-images of these functions.

Keywords

• Balanced function, minimal code, weakly regular plateaued function, weight distribution

References

1. A. Ashikhmin and A. Barg. Minimal vectors in lin- ear codes. IEEE Transactions on Information Theory, 44(5):2010–2017, 1998.
2. D. Bartoli and M. Bonini. Minimal linear codes in odd characteristic. IEEE Transactions on Information Theory, 65(7):4152–4155, 2019.
3. C. Carlet, C. Ding, and J. Yuan. Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Transactions on Information Theory, 51(6):2089–2102, 2005.
4. S. Chang and J. Y. Hyun. Linear codes from sim- plicial complexes. Designs, Codes and Cryptography, 86(10):2167–2181, 2018.
5. C. Ding. Linear codes from some 2-designs. IEEE Trans- actions on information theory, 61(6):3265–3275, 2015. [6] C. Ding.
6. A construction of binary linear codes
7. from Boolean functions. 339(9):2288–2303, 2016.
8. Discrete mathematics,

9. C. Ding, Z. Heng, and Z. Zhou. Minimal binary lin- ear codes. IEEE Transactions on Information Theory, 64(10):6536–6545, 2018.
10. C. Ding and J. Yuan. Covering and secret sharing with linear codes. DMTCS, 2731:11–25, 2003.
11. K. Ding and C. Ding. A class of two-weight and three- weight codes and their applications in secret sharing. IEEE Transactions on Information Theory, 61(11):5835–5842, 2015.
12. Z. Heng, C. Ding, and Z. Zhou. Minimal linear codes over finite fields. Finite Fields and Their Applications, 54:176–196, 2018.
13. W. C. Huffman, J.-L. Kim, and P. Solé. Linear codes from functions in Chapter 20 of Concise Encyclopedia of Coding Theory. Chapman and Hall/CRC, 2021.
14. N. Li and S. Mesnager. Recent results and problems on constructions of linear codes from cryptographic functions. Cryptography and Communications, 12:965–986, 2020.
15. R. Lidl and H. Niederreiter. Finite fields, volume 20. Cambridge university press, 1997.
16. J. L. Massey. Minimal codewords and secret sharing. In Proceedings of the 6th joint Swedish-Russian international workshop on information theory, pages 276–279, 1993.
17. S. Mesnager. Characterizations of plateaued and bent functions in characteristic p. In International Confer- ence on Sequences and Their Applications, pages 72–82. Springer, 2014.
18. S. Mesnager. Linear codes with few weights from weakly regular bent functions based on a generic construction. Cryptography and Communications, 9(1):71–84, 2017.
19. S. Mesnager, F. Özbudak, and A. Sınak. Linear codes from weakly regular plateaued functions and their secret sharing schemes. Designs, Codes and Cryptography, 87(2- 3):463–480, 2019.
20. S. Mesnager, Y. Qi, H. Ru, and C. Tang. Minimal linear codes from characteristic functions. IEEE Transactions on Information Theory, 2020.
21. S. Mesnager and A. Sınak. Infinite classes of six-weight linear codes derived from weakly regular plateaued func- tions. In 2020 International Conference on Information Security and Cryptology (ISCTURKEY), pages 93–100. IEEE, 2020.
22. S. Mesnager and A. Sınak. Several classes of minimal linear codes with few weights from weakly regular plateaued functions. IEEE Transactions on Information Theory, 66(4):2296–2310, 2020.
23. S. Mesnager, A. Sınak, and O. Yayla. Minimal linear codes with few weights and their secret sharing. International Journal of Information Security Science, 8(4):77–87, 2019.
24. Y. Qi, T. Yang, and B. Dai. Minimal linear codes from vectors with given weights. IEEE Communications Letters, 24(12):2674–2677, 2020.
25. O. S. Rothaus. On “bent” functions. Journal of Combi- natorial Theory, Series A, 20(3):300–305, 1976.
26. A. Sınak. Minimal linear codes from weakly regular plateaued balanced functions. 2020.
27. Discrete Mathematics,
28. C. Tang, N. Li, Y. Qi, Z. Zhou, and T. Helleseth. Linear codes with two or three weights from weakly regular bent functions. IEEE Transactions on Information Theory, 62(3):1166–1176, 2016.
29. G. Xu and L. Qu. Three classes of minimal linear codes over the finite fields of odd characteristic. IEEE Transactions on Information Theory, 65(11):7067–7078, 2019.
30. G. Xu, L. Qu, and X. Cao. Minimal linear codes from maiorana-mcfarland functions. Finite Fields and Their Applications, 65:101688, 2020.
31. G. Xu, L. Qu, and G. Luo. Minimal linear codes from weakly regular bent functions. The 11th International Conference on Sequences and Their Applications (SETA 2020).
32. Y. Zheng and X.-M. Zhang. Plateaued functions. In ICICS, volume 99, pages 284–300. Springer, 1999.