Define the bar product $C_{1} \mid C_{2}$ of binary linear codes $C_{1}$ and $C_{2}$, where $C_{2}$ is a subcode of $C_{1}$. Relate the rank and minimum distance of $C_{1} \mid C_{2}$ to those of $C_{1}$ and $C_{2}$ and justify your answer.

What is a parity check matrix for a linear code? If $C_{1}$ has parity check matrix $P_{1}$ and $C_{2}$ has parity check matrix $P_{2}$, find a parity check matrix for $C_{1} \mid C_{2}$.

Using the bar product construction, or otherwise, define the Reed-Muller code $R M(d, r)$ for $0 \leqslant r \leqslant d$. Compute the rank of $R M(d, r)$. Show that all but two codewords in $R M(d, 1)$ have the same weight. Given $d$, for which $r$ is it true that all elements of $R M(d, r)$ have even weight? Justify your answer.