  Let $G$ be a finite abelian group written additively, and let $r$ be a
multiple of its exponent. The modified Erd\H{o}s-Ginzburg-Ziv constant
$\mathsf{s}_r'(G)$ is the smallest integer $s$ such that every zero-sum
sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We
find exact values of $\mathsf{s}_{2k}'(\mathbb{Z}_2^d)$ for $d \leq 2k+1$.
