Describe the construction of the Reed-Miller code $R M(m, d)$. Establish its information rate and minimum weight.

Show that every codeword in $R M(d, d-1)$ has even weight. By considering $\mathbf{x} \wedge \mathbf{y}$ with $\mathbf{x} \in R M(m, r)$ and $\mathbf{y} \in R M(m, m-r-1)$, or otherwise, show that $R M(m, m-r-1) \subseteq R M(m, r)^{\perp}$. Show that, in fact, $R M(m, m-r-1)=R M(m, r)^{\perp} .$