An application of coding theory to estimating Davenport constants

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 ${\text{Dav}}_j(G)$ is defined as the smallest $\mathrm{\ell }$ such that each sequence over $G$ of length at least $\mathrm{\ell }$ 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.