= Solution
The <Reed-Muller code> is spanned by the evaluation <vectors> of the square-free monomials
$$
\prod_{i\in I}x_i,\qquad |I|\leq r,
$$
form a <basis>: every Boolean <function> has a unique algebraic normal form, so the monomials remain independent after evaluation on $\mathbb F_2^d$. Thus
$$
\dim\operatorname{RM}(d,r)=\sum_{j=0}^r\binom dj.
$$
Induction using the decomposition $(u,u+v)$, with $u\in\operatorname{RM}(d-1,r)$ and $v\in\operatorname{RM}(d-1,r-1)$, gives the lower bound $2^{d-r}$ for every nonzero weight. The monomial $x_1\cdots x_r$ attains it, so the minimum distance is
$$
\boxed{d_{\min}=2^{d-r}.}
$$
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