  For a finite abelian group $A$, define $f(A)$ to be the minimum integer such
that for every complete digraph $\Gamma$ on $f$ vertices and every map
$w:E(\Gamma) \rightarrow A$, there exists a directed cycle $C$ in $\Gamma$ such
that $\sum_{e \in E(C)}w(e) = 0$. The study of $f(A)$ was initiated by Alon and
Krivelevich (2021). In this article, we prove that $f(\mathbb{Z}_p^k) = O(pk
(\log k)^2)$, where $p$ is prime, with an improved bound of $O(k \log k)$ when
$p = 2$. These bounds are tight up to a factor which is polylogarithmic in $k$.
