Abstract
Linear complementary dual codes (shortly LCD codes) are codes whose intersections with their dual codes are trivial. These codes were first introduced by Massey in 1992. Nowadays, LCD codes are extensively studied in the literature and widely applied in data storage, cryptography, etc. In this paper, we prove some properties of binary LCD codes using their shortened and punctured codes. We also present some inequalities for the largest minimum weight \(d_{LCD}(n,k)\) of binary LCD [n, k] codes for given length n and dimension k. Furthermore, we give two tables with the values of \(d_{LCD}(n,k)\) for \(k\le 32\) and \(n\le 40\), and two tables with classification results.
Similar content being viewed by others
References
Araya M., Harada M.: On the minimum weights of binary linear complementary dual codes. Cryptogr. Commun. 12, 285–300 (2020).
Araya M., Harada M., Saito K.: Characterization and classification of optimal LCD codes. Des. Codes Cryptogr. 89, 617–640 (2021).
Bouyukliev I.: The Program Generation in the Software Package QextNewEdition. In: Bigatti A., Carette J., Davenport J., Joswig M., de Wolff T. (eds.) Mathematical Software - ICMS 2020, vol. 12097. Lecture Notes in Computer Science. Springer, Cham (2020).
Bouyuklieva S.: Binary optimal LCD codes [Data set], Zenodo, (2021). https://doi.org/10.5281/zenodo.5163242
Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10, 131–150 (2016).
Carlet C., Mesnager S., Tang C., Qi Y., Pellikaan R.: Linear codes over \(\mathbb{F}_q\) are equivalent to LCD codes for \(q > 3\). IEEE Trans. Inform. Theory 64, 3010–3017 (2018).
Carlet C., Mesnager S., Tang C., Qi Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory 65(1), 39–49 (2019).
Dougherty S.T., Kim J.-L., Ozkaya B., Sok L., Solé P.: The combinatorics of LCD codes: linear programming bound and orthogonal matrices. Int. J. Inf. Coding Theory 4, 116–128 (2017).
Galvez L., Kim J.-L., Lee N., Roe Y.G., Won B.-S.: Some bounds on binary LCD codes. Cryptogr. Commun. 10, 719–728 (2018).
Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de/. Accessed 22 Aug 2021.
Gulliver T.A., Östergård P.R.J.: Binary optimal linear rate \(1/2\) codes. Discret. Math. 283, 255–261 (2004).
Harada M., Saito K.: Binary linear complementary dual codes. Cryptogr. Commun. 11, 677–696 (2019).
Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).
Kaski P., Östergård P.R.: Classification Algorithms for Codes and Designs. Springer, New York (2006).
Massey J.L.: Linear codes with complementary duals. Discret. Math. 106(107), 337–342 (1992).
McKay B.: Isomorph-free exhaustive generation. J. Algorithms 26, 306–324 (1998).
Acknowledgements
I would like to thank Petar Boyvalenkov for focusing my attention to LCD codes and introducing me to the main literature related to this type of codes. I am also grateful to Iliya Bouyukliev for the included restriction about LCD codes in his program Generation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. D. Tonchev.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was supported in part by Bulgarian NSF contract KP-06-N32/2-2019.
Rights and permissions
About this article
Cite this article
Bouyuklieva, S. Optimal binary LCD codes. Des. Codes Cryptogr. 89, 2445–2461 (2021). https://doi.org/10.1007/s10623-021-00929-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-021-00929-w