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:
[1]

A. CanteautP. 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.

[2]

C. CarletP. 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.

[3]

J. Cannon, W. Bosma, C. Fieker and E. Stell, Handbook of Magma Functions, Version 2.19, Sydney, 2013.

[4]

X. CaoW. 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.

[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.

[6]

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

[7]

C. Ding, Designs from Iinear Codes, World Scientific, Singapore, 2019.

[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.

[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.

[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.

[11]

Z. HengC. 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.

[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.

[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.

[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.

[15] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge, 1997. 
[16]

J. Luo, On binary cyclic codes with five nonzero weights, arXiv: 0904.2237v1, 2009.

[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.

[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.

[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.

[20]

C. TangC. 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.

[21]

C. TangN. LiY. QiZ. 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.

[22]

X. WangD. Zheng and Y. Zhang, A class of subfield codes of linear codes and their duals, Cryptogr. Commun, 13 (2021), 173-196. 

[23]

X. WangD. ZhengL. 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.

[24]

Z. ZhouC. 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.

[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.

show all references

References:
[1]

A. CanteautP. 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.

[2]

C. CarletP. 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.

[3]

J. Cannon, W. Bosma, C. Fieker and E. Stell, Handbook of Magma Functions, Version 2.19, Sydney, 2013.

[4]

X. CaoW. 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.

[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.

[6]

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

[7]

C. Ding, Designs from Iinear Codes, World Scientific, Singapore, 2019.

[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.

[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.

[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.

[11]

Z. HengC. 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.

[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.

[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.

[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.

[15] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge, 1997. 
[16]

J. Luo, On binary cyclic codes with five nonzero weights, arXiv: 0904.2237v1, 2009.

[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.

[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.

[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.

[20]

C. TangC. 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.

[21]

C. TangN. LiY. QiZ. 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.

[22]

X. WangD. Zheng and Y. Zhang, A class of subfield codes of linear codes and their duals, Cryptogr. Commun, 13 (2021), 173-196. 

[23]

X. WangD. ZhengL. 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.

[24]

Z. ZhouC. 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.

[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.

Table 1.  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 $
Table 2.  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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