  Let $t \in \{2,8,10,12,14,16,18\}$ and $n=31s+t\geq 14$, $d_{a}(n,5)$ 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 $t\neq 16$, 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.
