On the largest minimum distances of [n,6] LCD codes (2406.02065v1)

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 $s \ge 0$ and $0\leq t\leq 62$ are two integers and $n=63s+t$. Using the theories of defining vectors, generalized anti-codes, reduced codes and nested codes, we exactly determine $d_{l}(n,6)$ for $t \notin{21,22,25,26,33,34,37,38,45,46}$, 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$t\in{25,33,45}$.