American Institute of Mathematical Sciences

Advanced
August  2018, 12(3): 579-594. doi: 10.3934/amc.2018034

Parameters of LCD BCH codes with two lengths

Haode Yan 1, , Hao Liu 2, , Chengju Li 3,, and  Shudi Yang 4,

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
2. 

Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
3. 

Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China
4. 

School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

* Corresponding author: Chengju Li

Received  August 2017 Revised  March 2018 Published  July 2018

Fund Project: The work was supported by the National Natural Science Foundation of China (NSFC) under Grant 11701179, the Shanghai Sailing Program under Grant 17YF1404300, the Foundation of Science and Technology on Information Assurance Laboratory under Grant KJ-17-007, the National NSFC under Grant 11701317, the Shanghai Natural Science Foundation under Grant 17ZR1408400, and the National Key R&D Program of China under Grant 2017YFB0802302.

In this paper, we study LCD BCH codes over the finite field GF$(q)$ with two types of lengths $n$, where $n = q^l+1$ and $n = (q^l+1)/(q+1)$. Several classes of LCD BCH codes are given and their parameters are determined or bounded by exploring the cyclotomic cosets modulo $n$. For $n = q^l+1$, we determine the dimensions of the codes with designed distance $δ$, where $q^{\lfloor\frac{l+1}{2}\rfloor}+1 ≤ δ ≤q ^{\lfloor\frac{l+3}{2}\rfloor}+1$. For $n = (q^l+1)/(q+1)$, the dimensions of the codes with designed distance $δ$ are presented, where $2 ≤ δ ≤q ^\frac{l-1}{2}+1$.

Keywords: LCD codes, BCH codes, cyclic codes, linear codes, reversible codes.
Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 94B15.
Citation: Haode Yan, Hao Liu, Chengju Li, Shudi Yang. Parameters of LCD BCH codes with two lengths. Advances in Mathematics of Communications, 2018, 12 (3) : 579-594. doi: 10.3934/amc.2018034
References:
[1]

S. A. Aly, A. Klappenecker and P. K. Sarvepalli, On quantum and classical BCH codes, IEEE Trans. Inf. Theory, 53 (2007), 1183-1188.  doi: 10.1109/TIT.2006.890730.

[2]

K. Boonniyoma and S. Jitman, Complementary dual subfield linear codes over finite fields, preprint, arXiv: 1605.06827.

[3]

C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, in Coding Theory and Applications (eds. R. Pinto, P. R. Malonek and P. Vettori), CIM Series in Mathematical Sciences, Springer Verlag, 3 (2014), 97-105.

[4]

C. Carlet, S. Mesnager, C. Tang and Y. Qi, Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr., (2018), 1-14.  doi: 10.1007/s10623-018-0463-8.

[5]

C. Carlet, S. Mesnager, C. Tang, Y. Qi and R. Pellikaan, Linear codes over $\Bbb F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inf. Theory, 64 (2018), 3010-3017.  doi: 10.1109/TIT.2018.2789347.

[6]

P. Charpin, Open problems on cyclic codes, in Handbook of Coding Theory (eds. V. S. Pless and W. C. Huffman), Amsterdam, The Netherlands: Elsevier, 1 (1998), 963-1063.

[7]

B. Chen and H. Liu, New constructions of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63 (2017), 2843-2847.  doi: 10.1109/TIT.2017.2748955.

[8]

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

[9]

C. Ding, X. Du and Z. Zhou, The Bose and minimum distance of a class of BCH codes, IEEE Trans. Inf. Theory, 61 (2015), 2351-2356.  doi: 10.1109/TIT.2015.2409838.

[10]

C. Ding, C. 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.

[11]

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

[12]

C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., (2017), 1-18.  doi: 10.1007/s10623-017-0447-0.

[13]

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

[14]

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

[15]

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

[16]

H. Liu, C. 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.

[17]

H. B. Mann, On the number of information symbols in Bose-Chaudhuri Codes, Inf. Control, 5 (1962), 153-162.  doi: 10.1016/S0019-9958(62)90298-X.

[18]

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

[19]

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.

[20]

D. Yue and Z. Hu, On the dimension and minimum distance of BCH codes over ${\rm{GF}}(q)$, J. Electronics (China), 13 (1996), 216-221. 

[21]

S. Zhu, B. Pang and Z. Sun, The reversible negacyclic codes over finite fields, preprint, arXiv: 1610.08206.

show all references

References:
[1]

S. A. Aly, A. Klappenecker and P. K. Sarvepalli, On quantum and classical BCH codes, IEEE Trans. Inf. Theory, 53 (2007), 1183-1188.  doi: 10.1109/TIT.2006.890730.

[2]

K. Boonniyoma and S. Jitman, Complementary dual subfield linear codes over finite fields, preprint, arXiv: 1605.06827.

[3]

C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, in Coding Theory and Applications (eds. R. Pinto, P. R. Malonek and P. Vettori), CIM Series in Mathematical Sciences, Springer Verlag, 3 (2014), 97-105.

[4]

C. Carlet, S. Mesnager, C. Tang and Y. Qi, Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr., (2018), 1-14.  doi: 10.1007/s10623-018-0463-8.

[5]

C. Carlet, S. Mesnager, C. Tang, Y. Qi and R. Pellikaan, Linear codes over $\Bbb F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inf. Theory, 64 (2018), 3010-3017.  doi: 10.1109/TIT.2018.2789347.

[6]

P. Charpin, Open problems on cyclic codes, in Handbook of Coding Theory (eds. V. S. Pless and W. C. Huffman), Amsterdam, The Netherlands: Elsevier, 1 (1998), 963-1063.

[7]

B. Chen and H. Liu, New constructions of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63 (2017), 2843-2847.  doi: 10.1109/TIT.2017.2748955.

[8]

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

[9]

C. Ding, X. Du and Z. Zhou, The Bose and minimum distance of a class of BCH codes, IEEE Trans. Inf. Theory, 61 (2015), 2351-2356.  doi: 10.1109/TIT.2015.2409838.

[10]

C. Ding, C. 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.

[11]

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

[12]

C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., (2017), 1-18.  doi: 10.1007/s10623-017-0447-0.

[13]

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

[14]

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

[15]

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

[16]

H. Liu, C. 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.

[17]

H. B. Mann, On the number of information symbols in Bose-Chaudhuri Codes, Inf. Control, 5 (1962), 153-162.  doi: 10.1016/S0019-9958(62)90298-X.

[18]

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

[19]

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.

[20]

D. Yue and Z. Hu, On the dimension and minimum distance of BCH codes over ${\rm{GF}}(q)$, J. Electronics (China), 13 (1996), 216-221. 

[21]

S. Zhu, B. Pang and Z. Sun, The reversible negacyclic codes over finite fields, preprint, arXiv: 1610.08206.

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