doi: 10.3934/amc.2021014

Ternary Primitive LCD BCH codes

1. 

Department of Mathematics, Jinling Institute of Technology, Nanjing, 211169, China

2. 

State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China

3. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211100, China

4. 

School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

5. 

Department of Mathematics, Nanjing Forestry University, Nanjing 210037, China

Received  December 2020 Published  May 2021

Fund Project: This paper was supported by the National Natural Science Foundation of China (No.61772015), the Foundation of Science and Technology on Information Assurance Laboratory (No.KJ-17-010) and the Foundation of Jinling Institute of Technology (No.JIT-B-202016, No.JIT-FHXM-2020). Y. Wu was sponsored by NUPTSF (No. NY220137). (Corresponding author:Yansheng Wu.)

Absolute coset leaders were first proposed by the authors which have advantages in constructing binary LCD BCH codes. As a continue work, in this paper we focus on ternary linear codes. Firstly, we find the largest, second largest, and third largest absolute coset leaders of ternary primitive BCH codes. Secondly, we present three classes of ternary primitive BCH codes and determine their weight distributions. Finally, we obtain some LCD BCH codes and calculate some weight distributions. However, the calculation of weight distributions of two of these codes is equivalent to that of Kloosterman sums.

Citation: Xinmei Huang, Qin Yue, Yansheng Wu, Xiaoping Shi. Ternary Primitive LCD BCH codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021014
References:
[1]

C. Carlet and S. Guilley, Complementary dual codes for countermeasures to side-channel attacks, Adv. Math. Commun., 10 (2016), 131-150.  doi: 10.3934/amc.2016.10.131.  Google Scholar

[2]

C. CarletS. MesnagerC. Tang and Y. Qi, Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr., 86 (2018), 2605-2618.  doi: 10.1007/s10623-018-0463-8.  Google Scholar

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C. CarletS. MesnagerC. TangY. Qi and R. Pellikaan, Linear codes over $\Bbb F_q$ which are equivalent to LCD codes, IEEE Trans. Inf. Theory, 64 (2018), 3010-3017.  doi: 10.1109/TIT.2018.2789347.  Google Scholar

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B. Chen and H. Liu, New constructions of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63 (2017), 2843-2847.   Google Scholar

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P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.  Google Scholar

[6]

C. Ding, Parameters of several classes of BCH codes, IEEE Trans. Inf. Theory, 61 (2015), 5322-5330.  doi: 10.1109/TIT.2015.2470251.  Google Scholar

[7]

C. DingC. Fan and Z. Zhou, The dimension and minimum distance of two classes of primitive BCH codes, Finite Fields Appl., 45 (2017), 237-263.  doi: 10.1016/j.ffa.2016.12.009.  Google Scholar

[8]

X. HuangQ. YueY. Wu and X. Shi, Binary Primitive LCD BCH codes, Des. Codes Cryptogr., 88 (2020), 2453-2473.  doi: 10.1007/s10623-020-00795-y.  Google Scholar

[9]

L. Jin, Construction of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63 (2017), 2843-2847.   Google Scholar

[10]

C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., 86 (2018), 2261-2278.  doi: 10.1007/s10623-017-0447-0.  Google Scholar

[11]

C. LiC. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63 (2017), 4344-4356.  doi: 10.1109/TIT.2017.2672961.  Google Scholar

[12]

C. LiP. Wu and F. Liu, On two classes of primitive BCH codes and some related codes, IEEE Trans. Inf. Theory, 65 (2019), 3830-3840.  doi: 10.1109/TIT.2018.2883615.  Google Scholar

[13]

C. LiQ. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.  Google Scholar

[14]

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc., 1983.  Google Scholar

[15]

F. LiQ. Yue and Y. Wu, Designed distances and parameters of new LCD BCH codes over finite fields, Cryptogr. Commun., 12 (2020), 147-163.  doi: 10.1007/s12095-019-00385-3.  Google Scholar

[16]

S. LiC. DingM. Xiong and G. Ge, Narrow-sense BCH codes over GF$(q)$ with length $n = \frac{q^m-1}{q-1}$, IEEE Trans. Inf. Theory, 63 (2017), 7219-7236.  doi: 10.1109/TIT.2017.2743687.  Google Scholar

[17]

S. LiC. LiC. Ding and H. Liu, Two families of LCD BCH codes, IEEE Trans. Inf. Theory, 63 (2017), 5699-5717.   Google Scholar

[18]

H. LiuC. Ding and C. Li, Dimensions of three types of BCH codes over GF$(q)$, Discrete Math., 340 (2017), 1910-1927.  doi: 10.1016/j.disc.2017.04.001.  Google Scholar

[19]

J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380.  doi: 10.1016/S0019-9958(64)90438-3.  Google Scholar

[20]

X. Shi, Q. Yue and S. Yang, New LCD MDS codes constructed from generalized Reed-Solomon codes, J. Algebra Appl., 18 (2018), 1950150. doi: 10.1142/S0219498819501500.  Google Scholar

[21]

K. K. Tzeng and C. R. P. Hartmann, On the minimum distance of certain reversible cyclic codes, IEEE Trans. Inf. Theory, 16 (1970), 644-646.  doi: 10.1109/tit.1970.1054517.  Google Scholar

[22]

Y. Wu and Q. Yue, Factorizations of binomial polynomials and enumerations of LCD and self-dual constacyclic codes, IEEE Trans. Inf. Theory, 65 (2019), 1740-1751.   Google Scholar

[23]

H. YanH. LiuC. Li and S. Yang, Parameters of LCD BCH codes with two lengths, Adv. Math. Commun., 12 (2018), 579-594.  doi: 10.3934/amc.2018034.  Google Scholar

[24]

X. Yang and J. L. Massey, The necessary and sufficient condition for a cyclic code to have a complementary dual, Discrete Math., 126 (1994), 391-393.  doi: 10.1016/0012-365X(94)90283-6.  Google Scholar

[25]

Z. ZhouX. LiC. Tang and C. Ding, Binary LCD codes and self-orthogonal codes from a generic construction, IEEE Trans. Inf. Theory, 65 (2019), 16-27.  doi: 10.1109/TIT.2018.2823704.  Google Scholar

show all references

References:
[1]

C. Carlet and S. Guilley, Complementary dual codes for countermeasures to side-channel attacks, Adv. Math. Commun., 10 (2016), 131-150.  doi: 10.3934/amc.2016.10.131.  Google Scholar

[2]

C. CarletS. MesnagerC. Tang and Y. Qi, Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr., 86 (2018), 2605-2618.  doi: 10.1007/s10623-018-0463-8.  Google Scholar

[3]

C. CarletS. MesnagerC. TangY. Qi and R. Pellikaan, Linear codes over $\Bbb F_q$ which are equivalent to LCD codes, IEEE Trans. Inf. Theory, 64 (2018), 3010-3017.  doi: 10.1109/TIT.2018.2789347.  Google Scholar

[4]

B. Chen and H. Liu, New constructions of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63 (2017), 2843-2847.   Google Scholar

[5]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.  Google Scholar

[6]

C. Ding, Parameters of several classes of BCH codes, IEEE Trans. Inf. Theory, 61 (2015), 5322-5330.  doi: 10.1109/TIT.2015.2470251.  Google Scholar

[7]

C. DingC. Fan and Z. Zhou, The dimension and minimum distance of two classes of primitive BCH codes, Finite Fields Appl., 45 (2017), 237-263.  doi: 10.1016/j.ffa.2016.12.009.  Google Scholar

[8]

X. HuangQ. YueY. Wu and X. Shi, Binary Primitive LCD BCH codes, Des. Codes Cryptogr., 88 (2020), 2453-2473.  doi: 10.1007/s10623-020-00795-y.  Google Scholar

[9]

L. Jin, Construction of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63 (2017), 2843-2847.   Google Scholar

[10]

C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., 86 (2018), 2261-2278.  doi: 10.1007/s10623-017-0447-0.  Google Scholar

[11]

C. LiC. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63 (2017), 4344-4356.  doi: 10.1109/TIT.2017.2672961.  Google Scholar

[12]

C. LiP. Wu and F. Liu, On two classes of primitive BCH codes and some related codes, IEEE Trans. Inf. Theory, 65 (2019), 3830-3840.  doi: 10.1109/TIT.2018.2883615.  Google Scholar

[13]

C. LiQ. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.  Google Scholar

[14]

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc., 1983.  Google Scholar

[15]

F. LiQ. Yue and Y. Wu, Designed distances and parameters of new LCD BCH codes over finite fields, Cryptogr. Commun., 12 (2020), 147-163.  doi: 10.1007/s12095-019-00385-3.  Google Scholar

[16]

S. LiC. DingM. Xiong and G. Ge, Narrow-sense BCH codes over GF$(q)$ with length $n = \frac{q^m-1}{q-1}$, IEEE Trans. Inf. Theory, 63 (2017), 7219-7236.  doi: 10.1109/TIT.2017.2743687.  Google Scholar

[17]

S. LiC. LiC. Ding and H. Liu, Two families of LCD BCH codes, IEEE Trans. Inf. Theory, 63 (2017), 5699-5717.   Google Scholar

[18]

H. LiuC. Ding and C. Li, Dimensions of three types of BCH codes over GF$(q)$, Discrete Math., 340 (2017), 1910-1927.  doi: 10.1016/j.disc.2017.04.001.  Google Scholar

[19]

J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380.  doi: 10.1016/S0019-9958(64)90438-3.  Google Scholar

[20]

X. Shi, Q. Yue and S. Yang, New LCD MDS codes constructed from generalized Reed-Solomon codes, J. Algebra Appl., 18 (2018), 1950150. doi: 10.1142/S0219498819501500.  Google Scholar

[21]

K. K. Tzeng and C. R. P. Hartmann, On the minimum distance of certain reversible cyclic codes, IEEE Trans. Inf. Theory, 16 (1970), 644-646.  doi: 10.1109/tit.1970.1054517.  Google Scholar

[22]

Y. Wu and Q. Yue, Factorizations of binomial polynomials and enumerations of LCD and self-dual constacyclic codes, IEEE Trans. Inf. Theory, 65 (2019), 1740-1751.   Google Scholar

[23]

H. YanH. LiuC. Li and S. Yang, Parameters of LCD BCH codes with two lengths, Adv. Math. Commun., 12 (2018), 579-594.  doi: 10.3934/amc.2018034.  Google Scholar

[24]

X. Yang and J. L. Massey, The necessary and sufficient condition for a cyclic code to have a complementary dual, Discrete Math., 126 (1994), 391-393.  doi: 10.1016/0012-365X(94)90283-6.  Google Scholar

[25]

Z. ZhouX. LiC. Tang and C. Ding, Binary LCD codes and self-orthogonal codes from a generic construction, IEEE Trans. Inf. Theory, 65 (2019), 16-27.  doi: 10.1109/TIT.2018.2823704.  Google Scholar

Weight Frequency
0 1
$ \frac{2}3\cdot(3^m-3^{\frac m2}) $ $ \frac{3^m-1}{2} $
$ \frac{2}3\cdot(3^m+3^{\frac m2}) $ $ \frac{3^m-1}{2} $
Weight Frequency
0 1
$ \frac{2}3\cdot(3^m-3^{\frac m2}) $ $ \frac{3^m-1}{2} $
$ \frac{2}3\cdot(3^m+3^{\frac m2}) $ $ \frac{3^m-1}{2} $
Weight Frequency
0 1
$ \frac{2}3 \cdot(3^m-3^{\frac{m}2}) $ $ \frac{3^m-1}2 $
$ \frac{2}3 \cdot(3^m+3^{\frac{m}2}) $ $ \frac{3^m-1}2 $
$ \frac{1}3(2\cdot3^m+3^\frac{m}2)-1 $ $ 3^m-1 $
$ \frac{1}3(2\cdot3^m-3^\frac{m}2)-1 $ $ 3^m-1 $
$ 3^m-1 $ 2
Weight Frequency
0 1
$ \frac{2}3 \cdot(3^m-3^{\frac{m}2}) $ $ \frac{3^m-1}2 $
$ \frac{2}3 \cdot(3^m+3^{\frac{m}2}) $ $ \frac{3^m-1}2 $
$ \frac{1}3(2\cdot3^m+3^\frac{m}2)-1 $ $ 3^m-1 $
$ \frac{1}3(2\cdot3^m-3^\frac{m}2)-1 $ $ 3^m-1 $
$ 3^m-1 $ 2
Weight Frequency
0 1
$ \frac12\cdot(3^m-1) $ 12
$ \frac34\cdot(3^m-1) $ 8
$ 3^m-1 $ 6
Weight Frequency
0 1
$ \frac12\cdot(3^m-1) $ 12
$ \frac34\cdot(3^m-1) $ 8
$ 3^m-1 $ 6
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