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 $n=31s+t\ge 14$ 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 $t\ne 16$ 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.