# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021023
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## Some subfield codes from MDS codes

 1 College of Mathematics and Informatics, South China Agricultural University, Guangzhou, Guangdong 510642, China 2 School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, Hubei 430079, China

* Corresponding author: Jinquan Luo

Received  November 2020 Revised  March 2021 Early access July 2021

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, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal.

Citation: Can Xiang, Jinquan Luo. Some subfield codes from MDS codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021023
##### References:
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show all references

##### References:
 [1] A. Canteaut, P. Charpin and H. Dobbertin, Weight divisibility of cyclic codes, highly nonlinear functions on $\Bbb F_{2^n}$, and crosscorrelation of maximum-length sequences, SIAM J. Discrete Math, 13 (2000), 105-138.  doi: 10.1137/S0895480198350057.  Google Scholar [2] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr, 15 (1998), 125-156.  doi: 10.1023/A:1008344232130.  Google Scholar [3] J. Cannon, W. Bosma, C. Fieker and E. Stell, Handbook of Magma Functions, Version 2.19, Sydney, 2013. Google Scholar [4] X. Cao, W. Chou and J. Gu, On the number of solutions of certain diagonal equations over finite fields, Finite Fields Their Appl, 42 (2016), 225-252.  doi: 10.1016/j.ffa.2016.08.003.  Google Scholar [5] C. Ding and Z. Heng, The subfield codes of ovoid codes, IEEE Trans. Inf. Theory, 65 (2019), 4715-4729.  doi: 10.1109/TIT.2019.2907276.  Google Scholar [6] C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.  Google Scholar [7] C. Ding, Designs from Iinear Codes, World Scientific, Singapore, 2019.  Google Scholar [8] C. Ding, A construction of binary linear codes from Boolean functions, Discret. Math., 339 (2016), 2288-2303.  doi: 10.1016/j.disc.2016.03.029.  Google Scholar [9] K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.  Google Scholar [10] Z. Heng and C. Ding, The subfield codes of hyperoval and conic codes, Finite Fields Their Appl, 56 (2019), 308-331.  doi: 10.1016/j.ffa.2018.12.006.  Google Scholar [11] Z. Heng, C. Ding and W. Wang, Optimal binary linear codes from maximal arcs, IEEE Trans. Inf. Theory, 66 (2020), 5387-5394.  doi: 10.1109/TIT.2020.2970405.  Google Scholar [12] Z. Heng and C. Ding, The subfield codes of ovoid vodes, IEEE Trans. Inform. Theory, 65 (2019), 4715-4729.  doi: 10.1109/TIT.2019.2907276.  Google Scholar [13] Z. Heng and Q. Yue, Complete weight distributions of two classes of cyclic codes, Cryptogr. Commun, 9 (2017), 323-343.  doi: 10.1007/s12095-015-0177-y.  Google Scholar [14] I. N. Landjev, Linear codes over finite fields and finite projective geometries, Discret. Math, 213 (2000), 211-244.  doi: 10.1016/S0012-365X(99)00183-1.  Google Scholar [15] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge, 1997.   Google Scholar [16] J. Luo, On binary cyclic codes with five nonzero weights, arXiv: 0904.2237v1, 2009. Google Scholar [17] S. Mesnager and A. Sinak, Several classes of minimal linear codes with few weights from weakly regular plateaued functions, IEEE Trans. Inf. Theory, 66 (2020), 2296-2310.  doi: 10.1109/TIT.2019.2956130.  Google Scholar [18] S. Mesnager, Linear codes with few weights from weakly regular bent functions based on a generic construction, Cryptogr. Commun, 9 (2017), 71-84.  doi: 10.1007/s12095-016-0186-5.  Google Scholar [19] X. Wang and D. Zheng, The subfield codes of several classes of linear codes, Cryptogr. Commun, 12 (2020), 1111-1131.  doi: 10.1007/s12095-020-00432-4.  Google Scholar [20] C. Tang, C. Xiang and K. Feng, Linear codes with few weights from inhomogeneous quadratic functions, Des. Codes Cryptogr, 83 (2017), 691-714.  doi: 10.1007/s10623-016-0267-7.  Google Scholar [21] C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.  Google Scholar [22] X. Wang, D. Zheng and Y. Zhang, A class of subfield codes of linear codes and their duals, Cryptogr. Commun, 13 (2021), 173-196.   Google Scholar [23] X. Wang, D. Zheng, L. Hu and X. Zeng, The weight distributions of two classes of binary cyclic codes, Finite Fields Their Appl, 34 (2015), 192-207.  doi: 10.1016/j.ffa.2015.01.012.  Google Scholar [24] Z. Zhou, C. Ding and J. Luo, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682.  doi: 10.1109/TIT.2013.2267722.  Google Scholar [25] Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl, 25 (2014) 79–93. doi: 10.1016/j.ffa.2013.08.005.  Google Scholar
The weight distribution of ${\mathcal{C}}_{ {\mathcal{M}}}^{(2)}$ for $m\geq 5$ odd
 Weight Multiplicity $0$ $1$ $2^{m}$ $1$ $2^{m-1}$ $9\cdot 2^{3 m-4} + 5 \cdot 2^{m-2} - 2^{2 m-2}-2$ $2^{m-1}+1$ $4^{m-2} (-2 + 9\cdot 2^m)$ $(2^m \pm 2^{\frac{m+1}{2}})/2$ $2^{m-3} (-16 + 5\cdot 2^{2 m} + 2^{2 + m})/3$ $(2^m \pm 2^{\frac{m+1}{2}})/2+1$ $2^{2 m-3} (2 + 5\cdot 2^m)/3$ $(2^m \pm 2^{\frac{m+3}{2}})/2$ $2^{ m-5} (-2 + 2^m)^2 /3$ $(2^m \pm 2^{\frac{m+3}{2}})/2+1$ $2^{2 m-5} (-2 + 2^m)/3$
 Weight Multiplicity $0$ $1$ $2^{m}$ $1$ $2^{m-1}$ $9\cdot 2^{3 m-4} + 5 \cdot 2^{m-2} - 2^{2 m-2}-2$ $2^{m-1}+1$ $4^{m-2} (-2 + 9\cdot 2^m)$ $(2^m \pm 2^{\frac{m+1}{2}})/2$ $2^{m-3} (-16 + 5\cdot 2^{2 m} + 2^{2 + m})/3$ $(2^m \pm 2^{\frac{m+1}{2}})/2+1$ $2^{2 m-3} (2 + 5\cdot 2^m)/3$ $(2^m \pm 2^{\frac{m+3}{2}})/2$ $2^{ m-5} (-2 + 2^m)^2 /3$ $(2^m \pm 2^{\frac{m+3}{2}})/2+1$ $2^{2 m-5} (-2 + 2^m)/3$
The weight distribution of ${\mathcal{C}}_{ {\mathcal{M}}}^{(2)}$ for $m\geq 6$ and $m\equiv 2 \pmod 4$
 Weight Multiplicity $0$ $1$ $2^{m-1}$ $-2 + 17 \cdot 2^{m-3} + 29\cdot 2^{ 3 m-6} - 2^{2m-5} \cdot 23$ $2^{m - 1} + 1$ $4^{m-3} (29\cdot 2^m -20)$ $(2^m \pm 2^{\frac{m}{2}})/2$ $2^m (-16 + 3 \cdot 4^m + 2^m \cdot 7) /15$ $(2^m \pm 2^{\frac{m}{2}})/2+1$ $4^m (1 + 2^m)/5$ $(2^m \pm 2^{\frac{m+2}{2}})/2$ $2^{2 m-5} (-10 + 7 \cdot 2^m) /3$ $(2^m\pm 2^{\frac{m+2}{2}})/2+1$ $2^{2 m-5} (-4 + 7 \cdot 2^m) /3$ $(2^m \pm 2^{\frac{m+4}{2}})/2$ $2^{m-7} (8 + 4^m + 2^{1 + m} (-3)) /15$ $(2^m \pm 2^{\frac{m+4}{2}})/2+1$ $2^{2 m-7} (-4 + 2^m) /15$ $2^m$ $1$
 Weight Multiplicity $0$ $1$ $2^{m-1}$ $-2 + 17 \cdot 2^{m-3} + 29\cdot 2^{ 3 m-6} - 2^{2m-5} \cdot 23$ $2^{m - 1} + 1$ $4^{m-3} (29\cdot 2^m -20)$ $(2^m \pm 2^{\frac{m}{2}})/2$ $2^m (-16 + 3 \cdot 4^m + 2^m \cdot 7) /15$ $(2^m \pm 2^{\frac{m}{2}})/2+1$ $4^m (1 + 2^m)/5$ $(2^m \pm 2^{\frac{m+2}{2}})/2$ $2^{2 m-5} (-10 + 7 \cdot 2^m) /3$ $(2^m\pm 2^{\frac{m+2}{2}})/2+1$ $2^{2 m-5} (-4 + 7 \cdot 2^m) /3$ $(2^m \pm 2^{\frac{m+4}{2}})/2$ $2^{m-7} (8 + 4^m + 2^{1 + m} (-3)) /15$ $(2^m \pm 2^{\frac{m+4}{2}})/2+1$ $2^{2 m-7} (-4 + 2^m) /15$ $2^m$ $1$
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