Let $ d_{a}(n, 5) $ and $ d_{l}(n, 5) $ be the minimum weights of optimal binary $ [n, 5] $ linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate $ d_{l}(n, 5) $ of some families of binary $ [n, 5] $ LCD codes when $ n = 31s+t\geq 14 $ with $ s $ integer and $ t \in\; \{2, 8, 10, 12, 14, 16, 18\} $. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary $ [n, 5, d_{a}(n, 5)] $ LCD codes is verified, and we further derive that $ d_{l}(n, 5) = d_{a}(n, 5)-1 $ for $ t\neq 16 $ and $ d_{l}(n, 5) = 16s+6 = d_{a}(n, 5)-2 $ for $ t = 16 $. Combining them with known results on optimal LCD codes, $ d_{l}(n, 5) $ of all $ [n, 5] $ LCD codes are completely determined.
Citation: Yang Liu, Ruihu Li, Qiang Fu, Hao Song. On the minimum distances of binary optimal LCD codes with dimension 5[J]. AIMS Mathematics, 2024, 9(7): 19137-19153. doi: 10.3934/math.2024933
Let $ d_{a}(n, 5) $ and $ d_{l}(n, 5) $ be the minimum weights of optimal binary $ [n, 5] $ linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate $ d_{l}(n, 5) $ of some families of binary $ [n, 5] $ LCD codes when $ n = 31s+t\geq 14 $ with $ s $ integer and $ t \in\; \{2, 8, 10, 12, 14, 16, 18\} $. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary $ [n, 5, d_{a}(n, 5)] $ LCD codes is verified, and we further derive that $ d_{l}(n, 5) = d_{a}(n, 5)-1 $ for $ t\neq 16 $ and $ d_{l}(n, 5) = 16s+6 = d_{a}(n, 5)-2 $ for $ t = 16 $. Combining them with known results on optimal LCD codes, $ d_{l}(n, 5) $ of all $ [n, 5] $ LCD codes are completely determined.
| [1] | W. C. Huffman, V. Pless, Fundamentals of error-correcting codes, New York: Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511807077 |
| [2] | E. F. Assmus Jr., J. D. Key, Affine and projective planes, Discrete Math., 83 (1990), 161–187. https://doi.org/10.1016/0012-365X(90)90003-Z |
| [3] |
L. Lu, R. Li, L. Guo, Q. Fu, Maximal entanglement entanglement-assisted quantum codes constructed from linear codes, Quantum Inf. Process., 14 (2015), 165–182. https://doi.org/10.1007/s11128-014-0830-y doi: 10.1007/s11128-014-0830-y
|
| [4] | J. L. Massey, Linear codes with complementary duals, Discrete Math., 106-107 (1992), 337–342. https://doi.org/10.1016/0012-365X(92)90563-U |
| [5] | C. Carlet, S. Guilley, Complementary dual codes for countermeasures to side-channel attacks, In: R. Pinto, P. Rocha Malonek, P. Vettori, Coding theory and applications, CIM Series in Mathematical Sciences, Berlin: Springer Verlag, 3 (2015), 97–105. https://doi.org/10.1007/978-3-319-17296-5_9 |
| [6] |
C. Carlet, S. Mesnager, C. Tang, Y. Qi, R. Pellikaan, Linear codes over $F_{q}$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inform. Theory, 64 (2018), 3010–3017. https://doi.org/10.1109/TIT.2018.2789347 doi: 10.1109/TIT.2018.2789347
|
| [7] |
L. Galvez, J. L. Kim, N. Lee, Y. G. Roe, B. S. Won, Some bounds on binary LCD codes, Cryptogr. Commun., 10 (2018), 719–728. https://doi.org/10.1007/s12095-017-0258-1 doi: 10.1007/s12095-017-0258-1
|
| [8] |
Q. Fu, R. Li, F. Fu, Y. Rao, On the construction of binary optimal LCD codes with short length, Int. J. Found. Comput. Sci., 30 (2019), 1237–1245. https://doi.org/10.1142/S0129054119500242 doi: 10.1142/S0129054119500242
|
| [9] |
M. Harada, K. Saito, Binary linear complementary dual codes, Cryptogr. Commun., 11 (2019), 677–696. https://doi.org/10.1007/s12095-018-0319-0 doi: 10.1007/s12095-018-0319-0
|
| [10] |
M. Araya, M. Harada, K. Saito, Characterization and classification of optimal LCD codes, Des., Codes Cryptogr., 89 (2021), 617–640. https://doi.org/10.1007/s10623-020-00834-8 doi: 10.1007/s10623-020-00834-8
|
| [11] |
M. Araya, M. Harada, On the minimum weights of binary linear complementary dual codes, Cryptogr. Commun., 12 (2020), 285–300. https://doi.org/10.1007/s12095-019-00402-5 doi: 10.1007/s12095-019-00402-5
|
| [12] |
M. Araya, M. Harada, K. Saito, On the minimum weights of binary LCD codes and ternary LCD codes, Finite Fields Appl., 76 (2021), 101925. https://doi.org/10.1016/j.ffa.2021.101925 doi: 10.1016/j.ffa.2021.101925
|
| [13] | S. Bouyuklieva, Optimal binary LCD codes, Des. Codes Cryptogr., 89 (2021), 2445–2461. https://doi.org/10.1007/s10623-021-00929-w |
| [14] | S. Li, M. Shi, Several constructions of optimal LCD codes over small finite fields, Cryptogr. Commun., 2024. https://doi.org/10.1007/s12095-024-00699-x |
| [15] | R. Li, Y. Liu, Q. Fu, On some problems of LCD codes, 2022 Symposium on Coding Theory and Cryptography and Their Related Topics, Shandong: Zi Bo, 2022. |
| [16] |
F. Li, Q. Yue, Y. Wu, Designed distances and parameters of new LCD BCH codes over finite fields, Cryptogr. Commun., 12 (2020), 147–163. https://doi.org/10.1007/s12095-019-00385-3 doi: 10.1007/s12095-019-00385-3
|
| [17] | W. C. Huffman, J. L. Kim, P. Solé, Concise encyclopedia of coding theory, Boca Raton: CRC Press, 2021,593–594. https://doi.org/10.1201/9781315147901 |
| [18] | M. Grassl, Code tables: bounds on the parameters of various types of codes. Available from: http://www.codetables.de/. |
| [19] |
R. Li, Z. Xu, X. Zhao, On the classification of binary optimal self-orthogonal codes, IEEE Trans. Inform. Theory, 54 (2008), 3778–3782. https://doi.org/10.1109/TIT.2008.926367 doi: 10.1109/TIT.2008.926367
|
| [20] | F. Zuo, R. Li, Y. Liu, Weight distributation of binary optimal codes and its application, 2012 International Conference on Computer Science and Information Processing (CSIP), Shaanxi: Xi'an, 2012,226–229. |
| [21] |
J. E. MacDonald, Design methods for maximum minimum-distance error-correcting codes, IBM J. Res. Dev., 4 (1960), 43–57. https://doi.org/10.1147/rd.41.0043 doi: 10.1147/rd.41.0043
|
| [22] | The MathWorks, MATLAB R2006a, Natick, MA, 2006. Available from: https://blogs.mathworks.com/steve/2006/03/06/mathworks-product-release-r2006a/. |
| [23] |
W. Bosma, J. Cannon, C. Playoust, The magma algebra system I: the user language, J. Symbolic Comput., 24 (1997), 235–265. https://doi.org/10.1006/jsco.1996.0125 doi: 10.1006/jsco.1996.0125
|
| [24] |
I. Bouyukliev, On the binary projective codes with dimension 6, Discrete Appl. Math., 154 (2006), 1693–1708. https://doi.org/10.1016/j.dam.2006.03.004 doi: 10.1016/j.dam.2006.03.004
|
Yang Liu, Ruihu Li, Qiang Fu, Hao Song. On the minimum distances of binary optimal LCD codes with dimension 5[J]. AIMS Mathematics, 2024, 9(7): 19137-19153. doi: 10.3934/math.2024933
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