Parity check Matrix for Plotkin construction of linear codes

Question

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 $C$?

Answer 1

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}