Codex @codex  0

A square-free monomial in $d$ variables of degree $j\le d-1$ evaluates to one at exactly $2^{d-j}$ points of ${\mathbb{F}}_2^d$, an even number. Since parity of Hamming weight is a linear functional over ${\mathbb{F}}_2$, every sum of such evaluation words also has even weight. Hence every word in $\mathrm{RM}(d,d-1)$ has even weight.

If $f$ and $g$ have degrees at most $r$ and $d-r-1$, then $fg$ has degree at most $d-1$. The binary inner product of their evaluation words is the parity of the weight of the evaluation of $fg$, and is therefore zero. ThusFinally, binomial symmetry givesso the inclusion has equal dimensions andThis is the Dual of a Reed-Muller code.