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A first step towards the skew duadic codes
Parameters of LCD BCH codes with two lengths
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 |
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 $2 ≤ δ ≤q ^\frac{l-1}{2}+1$.
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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