Construction methods for Galois LCD codes over finite fields. (English) Zbl 1542.94214
Summary: In this article, first we present a method for constructing many Hermitian LCD codes from a given Hermitian LCD code, and then provide several methods which utilize either a given \([n, k, d]\) linear code or a given \([n, k, d]\) Galois LCD code to construct new Galois LCD codes with different parameters. Using these construction methods, we construct several new \([n, k, d]\) ternary LCD codes with better parameters for \(26\le n\le 40,\) and \(21 \le k\le 30\). Also, optimal \(2\)-Galois LCD codes over \({\mathbb{F}}_{2^3}\) for code length, \(1\le n\le 15\) have been obtained. Finally, we extend some previously known results to the \(\sigma\)-inner product from Euclidean inner product.
References:
| [1] | Massey, JL, Linear codes with complementary duals, Discrete Math., 106-107, 337-342 (1992) · Zbl 0754.94009 · doi:10.1016/0012-365X(92)90563-U |
| [2] | Carlet, C.; Guilley, S., Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun., 10, 131-150 (2016) · Zbl 1352.94091 · doi:10.3934/amc.2016.10.131 |
| [3] | Guenda, K.; Jitman, S.; Gulliver, TA, Constructions of good entanglement-assisted quantum error correcting codes, Des. Codes Cryptogr., 86, 121-136 (2016) · Zbl 1387.81111 · doi:10.1007/s10623-017-0330-z |
| [4] | Liu, Z.; Wang, J., Linear complementary dual codes over rings, Des. Codes Cryptogr., 87, 3077-3086 (2019) · Zbl 1423.94154 · doi:10.1007/s10623-019-00664-3 |
| [5] | Liu, Z.; Wu, X., Notes on LCD codes over frobenius rings, IEEE Commun. Lett., 25, 361-364 (2021) · doi:10.1109/LCOMM.2020.3029073 |
| [6] | Liu, X.; Liu, H., LCD codes over finite chain rings, Finite Fields Their Appl., 34, 1-19 (2015) · Zbl 1392.94931 · doi:10.1016/j.ffa.2015.01.004 |
| [7] | Agrawal, A.; Verma, GK; Sharma, RK, Galois LCD codes over \(\mathbb{F}_q +u\mathbb{F}_q +v\mathbb{F}_q +uv\mathbb{F}_q\), Bull. Austral. Math. Soc., 107, 330-341 (2023) · Zbl 1529.94042 · doi:10.1017/S0004972722001344 |
| [8] | Ishizuka, K.; Saito, K., Construction for both self-dual codes and lcd codes, Adv. Math. Commun., 17, 1, 139-151 (2023) · Zbl 1521.94081 · doi:10.3934/amc.2021070 |
| [9] | Li, S., Shi, M.: Improved lower and upper bounds for LCD codes. arXiv:2206.04936 (2022) |
| [10] | Fan, Y.; Zhang, L., Galois self-dual constacyclic codes, Des. Codes Cryptogr., 84, 473-492 (2017) · Zbl 1381.94135 · doi:10.1007/s10623-016-0282-8 |
| [11] | Liu, X.; Fan, Y.; Liu, H., Galois LCD codes over finite fields, Finite Fields Their Appl., 49, 227-242 (2018) · Zbl 1392.94960 · doi:10.1016/j.ffa.2017.10.001 |
| [12] | Liu, H.; Pan, X., Galois hulls of linear codes over finite fields, Des. Codes Cryptogr., 88, 241-255 (2020) · Zbl 1452.94116 · doi:10.1007/s10623-019-00681-2 |
| [13] | Debnath, I.; Prakash, O.; Islam, H., Galois hulls of constacyclic codes over finite fields, Cryptogr. Commun., 15, 111-127 (2022) · Zbl 1518.94136 · doi:10.1007/s12095-022-00591-6 |
| [14] | Carlet, C.; Mesnager, S.; Tang, C.; Qi, Y., On \(\sigma \) -lcd codes, IEEE Trans. Inf. Theory, 65, 3, 1694-1704 (2019) · Zbl 1431.94181 · doi:10.1109/TIT.2018.2873130 |
| [15] | Liu, X., Liu, H., Yu, L.: New EAQEC codes constructed from galois lcd codes. Quantum Inf. Process. 19 (2019) · Zbl 1508.81593 |
| [16] | Hu, P., Liu, X.: Three classes of new EAQEC MDS codes. Quantum Inf. Process. 20 (2021) · Zbl 1509.81336 |
| [17] | Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235-265 (1997). doi:10.1006/jsco.1996.0125 · Zbl 0898.68039 |
| [18] | Talbi, S.; Batoul, A.; Tabue, AF; Mart’inez-Moro, E., Galois hulls of cyclic serial codes over a finite chain ring, Finite Fields Their Appl., 77 (2022) · Zbl 1476.14055 · doi:10.1016/j.ffa.2021.101950 |
| [19] | Thipworawimon, S.; Jitman, S., Hulls of linear codes revisited with applications, J. Appl. Math. Comput., 62, 325-340 (2019) · Zbl 1508.94074 · doi:10.1007/s12190-019-01286-7 |
| [20] | Cao, M., MDS codes with galois hulls of arbitrary dimensions and the related entanglement-assisted quantum error correction, IEEE Trans. Inf. Theory, 67, 7964-7984 (2021) · Zbl 1489.94185 · doi:10.1109/TIT.2021.3117562 |
| [21] | Dougherty, ST; Korban, A.; Şahinkaya, S., Self-dual additive codes, Appl. Algebra Eng. Commun. Comput., 33, 5, 569-586 (2022) · Zbl 1526.94056 · doi:10.1007/s00200-020-00473-5 |
| [22] | Araya, M.; Harada, M.; Saito, K., On the minimum weights of binary LCD codes and ternary LCD codes, Finite Fields Appl, 76 (2021) · Zbl 1492.94199 · doi:10.1016/j.ffa.2021.101925 |
| [23] | Araya, M.; Harada, M., On the classification of linear complementary dual codes, Discrete Math., 342, 1, 270-278 (2019) · Zbl 1417.94100 · doi:10.1016/j.disc.2018.09.034 |
| [24] | Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at. http://www.codetables.de · Zbl 1145.94300 |
| [25] | Harada, M., Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes, Des. Codes Cryptogr., 89, 10, 2295-2312 (2021) · Zbl 1483.94068 · doi:10.1007/s10623-021-00916-1 |
| [26] | Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., Wassermann, A.: Error-correcting linear codes: classification by isometry and applications (2006) · Zbl 1102.94001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
