Minimum distance of a bar product
= Minimum distance of a bar product
If $C_2\subseteq C_1$ have minimum distances $d_2,d_1$, respectively, then
$$
d(C_1|C_2)=\min(2d_1,d_2).
$$
Indeed, $\operatorname{wt}(u)+\operatorname{wt}(u+v)\geq\operatorname{wt}(v)$, while the words $(u,u)$ and $(0,v)$ attain the two candidate bounds.