Bar product of binary linear codes ($C_1|C_2$)

For length-$n$ binary linear codes $C_2\subseteq C_1$, their bar product isIt has length $2n$ and dimension $\dim C_1+\dim C_2$.

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.

If $H_i$ is a parity-check matrix of $C_i$, then a parity-check matrix of $C_1|C_2$ isIts equations say that the first half $u$ lies in $C_1$ and that the sum of the two halves lies in $C_2$.

For $0<r<d$,Consequently