  Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The
generalized Erd\H{o}s-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer
$s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence
of length $r$. We find exact values of $s_{2m}(\mathbb{Z}_2^d)$ for $d \leq
2m+1$. Connections to linear binary codes of maximal length and codes without a
forbidden weight are discussed.
