= Solution
A <linear code> of length $n$ over $\mathbb F_q$ is a linear subspace $C\subseteq\mathbb F_q^n$. A <parity-check matrix> $P$ satisfies $C=\ker P$. Its <minimum distance of a code> is the least <Hamming distance> between distinct codewords, equivalently the least <Hamming weight> of a nonzero word, because differences remain in the code.
For the usual binary <bar product of binary linear codes>, take equal-length binary codes with $C_2\subseteq C_1$ and define
$$
C_1|C_2=\{(u,u+v):u\in C_1,\ v\in C_2\}.
$$
It is linear, of length $2n$ and dimension $\dim C_1+\dim C_2$, since the parametrization is injective. If $v=0$ and the word is nonzero, its weight is $2\operatorname{wt}(u)\ge2d(C_1)$. If $v\ne0$, coordinatewise $\operatorname{wt}(u)+\operatorname{wt}(u+v)\ge\operatorname{wt}(v)\ge d(C_2)$. The words $(u,u)$ with $u$ of minimum nonzero weight and $(0,v)$ with $v$ of minimum weight achieve the two candidate bounds. Thus
$$
\boxed{d(C_1|C_2)=\min\{2d(C_1),d(C_2)\}.}
$$
For a word $(x,y)$, membership is equivalent to $P_1x=0$ and $P_2(x+y)=0$. Consequently a <parity-check matrix of a bar product> is
$$
\boxed{\begin{pmatrix}P_1&0\\P_2&P_2\end{pmatrix}.}
$$
The same distance argument actually needs only equal block lengths and a common field, not nesting. Over a general field replace the second row block by $(-P_2,P_2)$, so it tests $y-x\in C_2$. The displayed binary construction uses the conventional nesting.
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