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Let $G$ be a finite abelian group written additively, and let $r$ be amultiple 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-sumsequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. Wefind exact values of $\mathsf{s}_{2k}'(\mathbb{Z}_2^d)$ for $d \leq 2k+1$.

Modified Erd\H{o}s-Ginzburg-Ziv constants for $\mathbb{Z}_2^d$