Describe the construction of the Reed-Miller code RM(m,d). Establish its information rate and minimum weight.
Show that every codeword in RM(d,d−1) has even weight. By considering x∧y with x∈RM(m,r) and y∈RM(m,m−r−1), or otherwise, show that RM(m,m−r−1)⊆RM(m,r)⊥. Show that, in fact, RM(m,m−r−1)=RM(m,r)⊥.