Parity check Matrix for Plotkin construction of linear codes Let's say I have 2 linear codes, $C_1 = [n,k_1]$ and $C_2 = [n,k_2]$, and I have the parity check matricies $H_1,H_2$ for them. I use the Plotkin construction to create the code $C$ out of them (for every $u\in C_1$, $v\in C_2$, $(u|u+v)\in C$). How can I construct the parity check matrix $H$ of $C$?
 A: The $k\times n$ generator matrix $\mathbf G$ and the $(n-k)\times n$
parity-check matrix $\mathbf H$ of a $[n,k,d]$ linear code satisfy 
$$\mathbf{GH}^T = \mathbf 0.$$
The Plotkin construction $[u \mid u+v]$ where 
$u \in \mathcal C[n,k_1,d_1]$ and $v \in \mathcal C[n,k_2,d_2]$ gives
a $[2n,k_1+k_2]$ code whose $2n\times (k_1+k_2)$generator matrix is
$$\mathbf G = \left[\begin{matrix}
\mathbf G_1 & \mathbf G_1\\\mathbf 0&\mathbf G_2\end{matrix}\right]$$
and whose parity-check matrix is
$$\mathbf H = \left[\begin{matrix}
\mathbf H_1 & \mathbf 0\\\mathbf H_2&-\mathbf H_2\end{matrix}\right].$$
Note that
\begin{align}\mathbf{GH}^T &= \left[\begin{matrix}
\mathbf G_1 & \mathbf G_1\\\mathbf 0&\mathbf G_2\end{matrix}\right]
\left[\begin{matrix}
\mathbf H_1 & \mathbf 0\\\mathbf H_2&-\mathbf H_2\end{matrix}\right]^T\\
&= \left[\begin{matrix}
\mathbf G_1 & \mathbf G_1\\\mathbf 0&\mathbf G_2\end{matrix}\right]
\left[\begin{matrix}
\mathbf H_1^T & \mathbf H_2^T\\\mathbf 0&-\mathbf H_2^T\end{matrix}\right]\\
&= \left[\begin{matrix}
\mathbf G_1\mathbf H_1^T & \mathbf G_1\mathbf H_2^T-\mathbf G_1\mathbf H_2^T\\
\mathbf 0&-\mathbf G_2\mathbf H_2^T\end{matrix}\right]\\
&= \large{\mathbf 0}.
\end{align}
