On the minimum distances of binary optimal LCD codes with dimension 5

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