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A class of subfield codes of linear codes and their duals

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Abstract

Recently, subfield codes of some optimal linear codes have been studied. In this paper, we further investigate a class of subfield codes and generalize the results of the subfield codes of the conic codes in Ding and Wang (Finite Fields Appl. 56, 308–331, 2020). The weight distributions of these subfield codes and the parameters of their duals are determined. Some of the presented codes are optimal or almost optimal according to Grassl (2020) and their duals are distance-optimal with respect to the Sphere Packing bound if p > 3. As a byproduct, we directly obtain the weight distributions of the punctured codes, which is the same with the results presented in Du et al. (2019a, b), and determine the parameters of the duals of the punctured codes. These dual codes are distance-optimal with respect to the Sphere Packing bound with rare exceptions.

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References

  1. Cannon, J., Bosma, W., Fieker, C., Stell, E.: Handbook of Magma Functions. Version 2.19, Sydney (2013)

  2. 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)

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

  4. Coulter, R.S.: Further evaluations of Weil sums. Acta Arith. 86, 217–226 (1998)

  5. Coulter, R.S.: Explicit evaluations of some Weil sums. Acta Arith. 83, 241–251 (1998)

  6. Coulter, R.S.: The number of rational points of a class of Artin–Schreier curves. Finite Fields Appl. 8, 397–413 (2002)

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

  8. Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59(9), 5898–5904 (2013)

  9. Ding, C.: Linear codes from some 2–designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015)

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

  11. Ding, K., Ding, C.: Binary linear codes with three weights. IEEE Commun. Lett. 18(11), 1879–1882 (2014)

  12. 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)

  13. Dinh, H.Q., Li, C., Yue, Q.: Recent progress on weight distributions of cyclic codes over finite fields. J. Algebra Comb. Disc. Struc. Appl. 2, 39–63 (2015)

  14. Draper, S., Hou, X.: Explicit evaluation of certain exponential sums of quadratic functions over \(\mathbb {F}_{p^{n}}\), p odd. arXiv:0708.3619v1 (2007)

  15. Du, X., Wang, R., Fan, C.: Infinite families of 2–designs from a class of cyclic codes with two non–zeros. arxiv:1904.04242v1 (2019a)

  16. Du, X., Wang, R., Tang, C., Wang, Q.: Infinite families of 2–designs from two classes of linear codes. arxiv:1904.04242v1 (2019b)

  17. EI Rouayheb, S.Y., Georghiades, C.N., Soljanin, E., Sprintson, A.: Bounds on codes based on graph theory. IEEE Int. Symp. on Information Theory. Nice, France, pp. 1876–1879 (2007)

  18. Fan, J.: Optimal p–ary cyclic codes with minimum distance four. arXiv:1706.09188v2 (2018)

  19. Fan, C., Li, N., Zhou, Z.: A class of optimal ternary cyclic codes and their duals. Finite Fields Appl. 37, 193–202 (2016)

  20. Grassl, M.: Bounds on the minimum distance of linear codes. available online at http://www.codetables.de

  21. Han, D., Yan, H.: On an open problem about a class of optimal ternary cyclic codes. Finite Fields Appl. 59, 335–343 (2019)

  22. Heng, Z., Ding, C.: The subfield codes of Hyperoval and Conic codes. Finite Fields Appl. 56, 308–331 (2019)

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

  24. 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

  25. Kløve, T.: Codes for Error Detection. World Scientific, Hackensack (2007)

  26. Liu, H., Wang, X., Zheng, D.: On the weight distributions of a class of cyclic codes. Discrete Math. 341, 759–771 (2018)

  27. Li, N., Li, C., Helleseth, T., Ding, C., Tang, X.: Optimal ternary cyclic codes with minimum distance four and five. Finite Fields Appl. 30, 100–120 (2014)

  28. Li, N., Zhou, Z., Helleseth, T.: On a conjecture about a class of optimal ternary cyclic codes. Seventh International Workshop on Signal Design and its Applications in Communications (IWSDA). https://doi.org/10.1109/IWSDA.2015.7458415 (2015)

  29. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics, vol. 20, Cambridge University Press, Cambridge (1983)

  30. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error–Correcting Codes. North–Holland Publishing Company (1997)

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

  32. Xu, G., Cao, X., Xu, S.: Optimal p–ary cyclic codes with minimum distance four from monomials. Cryptogr. Commun. 8(4), 541–554 (2016)

  33. Zhou, Y., Kai, X., Zhu, S., Li, J.: On the minimum distance of negacyclic codes with two zeros. Finite Fields Appl. 55, 143–150 (2019)

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Acknowledgments

The authors sincerely thank the reviewers and the editor for their helpful comments and valuable suggestions, which have improved the presentation of this paper. This work was partially supported by National Natural Science Foundation of China under Grants 11971156, 12001175 and 61977021, and Hubei province science and technology innovation major project under Grant 2019ACA144.

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Authors and Affiliations

  1. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Xiaoqiang Wang & Dabin Zheng

  2. School of Computer Science and Information Engineering, Hubei University, Wuhan, 430062, China

    Yan Zhang

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  1. Xiaoqiang Wang
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  2. Dabin Zheng
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  3. Yan Zhang
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Correspondence to Dabin Zheng.

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Wang, X., Zheng, D. & Zhang, Y. A class of subfield codes of linear codes and their duals. Cryptogr. Commun. 13, 173–196 (2021). https://doi.org/10.1007/s12095-020-00460-0

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