Skip to main content
Log in

Two families of subfield codes with a few weights

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, two families of binary subfield codes with a few weights are presented from two special classes of linear codes, and their parameters are explicitly determined. Moreover, the parameters of the duals of these subfield codes are also studied. The two infinite families of subfield codes presented in this paper are distance-optimal with respect to the Griesmer bound and their duals are almost distance-optimal with respect to the sphere-packing bound.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Canteaut, A., Charpin, P., Dobbertin, H.: Weight divisibility of cyclic codes, highly nonlinear functions on \(\mathbb {F}_{2^{n}}\), and crosscorrelation of maximum-length sequences. SIAM Disc. Math. 13(1), 105–138 (2000)

    Article  Google Scholar 

  2. Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)

    Article  MathSciNet  Google Scholar 

  3. Cannon, J., Bosma, W., Fieker, C., Stell, E.: Handbook of magma functions, version 2.19. Sydney (2013)

  4. Ding, C., Heng, Z.: The subfield codes of ovoid codes. IEEE Trans. Inf. Theory 65(8), 4715–4729 (2019)

    Article  MathSciNet  Google Scholar 

  5. Ding, C.: Designs from Linear Codes. World Scientific, Singapore (2018)

    Book  Google Scholar 

  6. Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 60(6), 3265–3275 (2015)

    Article  MathSciNet  Google Scholar 

  7. Ding, C.: A construction of binary linear codes from Boolean functions. Discret. Math. 339(9), 2288–2303 (2016)

    Article  MathSciNet  Google Scholar 

  8. Ding, C., Li, C., Li, N., Zhou, Z.: Three-weight cyclic codes and their weight distributions. Discret. Math. 339(2), 415–427 (2016)

    Article  MathSciNet  Google Scholar 

  9. 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)

    Article  MathSciNet  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. Heng, Z., Ding, C.: The subfield codes of hyperoval and conic codes. Finite Fields Their Appl. 56, 308–331 (2019)

    Article  MathSciNet  Google Scholar 

  12. Heng, Z., Ding, C., Wang, W.: Optimal binary linear codes from maximal arcs. IEEE Trans. Inf. Theory 66(9), 5387–5394 (2020)

    Article  MathSciNet  Google Scholar 

  13. Heng, Z.: C.ding The subfield codes of [q + 1, 2,q] MDS codes Arxiv:2008.00695v2 (2020)

  14. Heng, Z., Wang, Q., Ding, C.: Two families of optimal linear codes and their subfield codes. IEEE Trans. Inf. Theory. https://doi.org/10.1109/TIT.2020.3006846

  15. Heng, Z., Yue, Q.: Complete weight distributions of two classes of cyclic codes. Cryptogr. Commun. 9(3), 323–343 (2017)

    Article  MathSciNet  Google Scholar 

  16. Griesmer, J. H.: A bound for error-correcting codes. IBM J. Res. Develop. 4(5), 532–542 (1960)

    Article  MathSciNet  Google Scholar 

  17. Huffman, W. C., Pless, V.: Fundamentals of error-correcting codes. Cambridge Univ Press, Cambridge (2003)

  18. Li, C., Yue, Q., Li, F.: Weight distributions of cyclic codes with respect to pairwise coprime order elements. Finite Fields Their Appl. 28, 94–114 (2014)

    Article  MathSciNet  Google Scholar 

  19. Li, C., Yue, Q.: Weight distributions of two classes of cyclic codes with respect to two distinct order elements. IEEE Trans. Inf. Theory 60(1), 296–303 (2014)

    Article  MathSciNet  Google Scholar 

  20. Klove, T.: Codes for Error Detection. World Scientific, Hackensack (2007)

    Book  Google Scholar 

  21. Lidl, R., Niederreiter, H.: Finite Fields. Cambridge Univ Press, Cambridge (1997)

    MATH  Google Scholar 

  22. Mesnager, S., Sinak, A.: Several classes of minimal linear codes with few weights from weakly regular plateaued functions. IEEE Trans. Inf. Theory 66(4), 2296–2310 (2020)

    Article  MathSciNet  Google Scholar 

  23. Mesnager, S.: Linear codes with few weights from weakly regular bent functions based on a generic construction. Cryptogr. Commun. 9(1), 71–84 (2017)

    Article  MathSciNet  Google Scholar 

  24. Tang, C., Xiang, C., Feng, K.: Linear codes with few weights from inhomogeneous quadratic functions. Des. Codes Cryptogr. 83(3), 691–714 (2017)

    Article  MathSciNet  Google Scholar 

  25. 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)

    Article  MathSciNet  Google Scholar 

  26. Wang, X., Zheng, D.: The subfield codes of several classes of linear codes, Cryptogr. Commun., https://doi.org/10.1007/s12095-020-00432-4 (2020)

  27. Zhou, Z., Ding, C., Luo, J., et al.: And A family of five-weight cyclic codes and their weight enumerators. IEEE Trans. Inf. Theory 59(10), 6674–6682 (2013)

    Article  MathSciNet  Google Scholar 

  28. Zhou, Z., Ding, C.: Seven classes of three-weight cyclic codes. IEEE Trans. Commun. 61(10), 4120–4126 (2013)

    Article  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the reviewers and the Editor, for their comments and suggestions that improved the presentation and quality of this paper. This paper was supported by the National Natural Science Foundation of China under grant numbers 11701187 and 11971175.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Can Xiang.

Additional information

Publisher’s note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiang, C., Yin, W. Two families of subfield codes with a few weights. Cryptogr. Commun. 13, 117–127 (2021). https://doi.org/10.1007/s12095-020-00457-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-020-00457-9

Keywords

Mathematics Subject Classification (2010)

Navigation