 Minimum distance of a bar product

If $C_2\subseteq C_1$ have minimum distances $d_2,d_1$, respectively, thenIndeed, $\mathrm{wt}(u)+\mathrm{wt}(u+v)\ge \mathrm{wt}(v)$, while the words $(u,u)$ and $(0,v)$ attain the two candidate bounds.