Minimum distances of binary optimal LCD codes of dimension five are completely determined

Let $t\in \{2,8,10,12,14,16,18\}$ and $n=31s+t\ge 14$, $d_a(n,5)$ and $d_l(n,5)$ be distances of binary $[n,5]$ optimal linear codes and optimal linear complementary dual (LCD) codes, respectively. We show that an $[n,5,d_a(n,5)]$ optimal linear code is not an LCD code, there is an $[n,5,d_l(n,5)]=[n,5,d_a(n,5)-1]$ optimal LCD code if $t\mathrm{\ne }16$, and an optimal $[n,5,d_l(n,5)]$ optimal LCD code has $d_l(n,5)=16s+6=d_a(n,5)-2$ for $t=16$. Combined with known results on optimal LCD code, $d_l(n,5)$ of all $[n,5]$ LCD codes are completely determined.