Scholar - SciOpen

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 $n = 31 s + t \geq 14$ 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 $d_{l} (n, 5) = 16 s + 6 = d_{a} (n, 5) - 2$ 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.