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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.   Google Scholar

[21]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.   Google Scholar

[21]

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

[1]

Xinmei Huang, Qin Yue, Yansheng Wu, Xiaoping Shi. Ternary Primitive LCD BCH codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021014
[2]

Angelot Behajaina. On BCH split metacyclic codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021045
[3]

José Joaquín Bernal, Diana H. Bueno-Carreño, Juan Jacobo Simón. Cyclic and BCH codes whose minimum distance equals their maximum BCH bound. Advances in Mathematics of Communications, 2016, 10 (2) : 459-474. doi: 10.3934/amc.2016018
[4]

Ranya Djihad Boulanouar, Aicha Batoul, Delphine Boucher. An overview on skew constacyclic codes and their subclass of LCD codes. Advances in Mathematics of Communications, 2021, 15 (4) : 611-632. doi: 10.3934/amc.2020085
[5]

Keita Ishizuka, Ken Saito. Construction for both self-dual codes and LCD codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021070
[6]

Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[7]

Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004
[8]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177
[9]

Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Advances in Mathematics of Communications, 2010, 4 (1) : 69-81. doi: 10.3934/amc.2010.4.69
[10]

Gustavo Terra Bastos, Reginaldo Palazzo Júnior, Marinês Guerreiro. Abelian non-cyclic orbit codes and multishot subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 631-650. doi: 10.3934/amc.2020035
[11]

Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41
[12]

Yunwen Liu, Longjiang Qu, Chao Li. New constructions of systematic authentication codes from three classes of cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 1-16. doi: 10.3934/amc.2018001
[13]

Sara D. Cardell, Joan-Josep Climent, Daniel Panario, Brett Stevens. A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes. Advances in Mathematics of Communications, 2020, 14 (3) : 437-453. doi: 10.3934/amc.2020047
[14]

Ettore Fornasini, Telma Pinho, Raquel Pinto, Paula Rocha. Composition codes. Advances in Mathematics of Communications, 2016, 10 (1) : 163-177. doi: 10.3934/amc.2016.10.163
[15]

Fernando Hernando, Tom Høholdt, Diego Ruano. List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes. Advances in Mathematics of Communications, 2012, 6 (3) : 259-272. doi: 10.3934/amc.2012.6.259
[16]

Liqin Hu, Qin Yue, Fengmei Liu. Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3). Advances in Mathematics of Communications, 2014, 8 (3) : 297-312. doi: 10.3934/amc.2014.8.297
[17]

Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039
[18]

Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225
[19]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55
[20]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (447)
  • HTML views (241)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy