On the largest minimum distances of [n,6] LCD codes

Yang Liu; Ruihu Li

doi:10.1016/j.heliyon.2024.e38238

paper

On the largest minimum distances of [n,6] LCD codes

arXiv:2406.02065 · doi:10.1016/j.heliyon.2024.e38238

Abstract

Linear complementary dual (LCD) codes can be used to against side-channel attacks and fault noninvasive attacks. Let $d_{a} (n, 6)$ and $d_{l} (n, 6)$ be the minimum weights of all binary optimal linear codes and LCD codes with length $n$ and dimension 6, respectively.In this article, we aim to obtain the values of $d_{l} (n, 6)$ for $n \geq 51$ by investigating the nonexistence and constructions of LCD codes with given parameters. Suppose that $s \geq 0$ and $0 \leq t \leq 62$ are two integers and $n = 63 s + t$ . Using the theories of defining vectors, generalized anti-codes, reduced codes and nested codes, we exactly determine $d_{l} (n, 6)$ for $t \in / {21, 22, 25, 26, 33, 34, 37, 38, 45, 46}$ , while we show that $d_{l} (n, 6) \in$ ${d_{a} (n, 6)$ $- 1, d_{a} (n, 6)}$ for $t \in {21, 22, 26, 34, 37, 38, 46}$ and $d_{l} (n, 6) \in$ ${d_{a} (n, 6) - 2,$ $d_{a} (n, 6) - 1}$ for$t\in{25,33,45\}$.

optimal linear code, LCD code,generalized anti-code, defining vector, reduced code