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 D_j(G) is defined as the smallest l such that each sequence over G of length at least l 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.