  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 show that $s_{2m}(\mathbb{Z}_2^d) \leq C_m 2^{d/m} + O(1)$
when $d\rightarrow\infty$, and $s_{2m}(\mathbb{Z}_2^d) \geq 2^{d/m} + 2m-1$
when $d=km$. We use results on $s_r(G)$ to prove new bounds for the codegree
Tur\'{a}n density of complete $r$-graphs.
