  We investigate a certain well-established generalization of the Davenport
constant. For $j$ a positive integer (the case $j=1$, is the classical one) and
a finite Abelian group $(G,+,0)$, the invariant $\Dav_j(G)$ is defined as the
smallest $\ell$ such that each sequence over $G$ of length at least $\ell$ has
$j$ disjoint non-empty zero-sum subsequences. We investigate these quantities
for elementary $2$-groups of large rank (relative to $j$). Using tools from
coding theory, we give fairly precise estimates for these quantities. We use
our results to give improved bounds for the classical Davenport constant of
certain groups.
