Relaxations of Hall's condition: Optimal batch codes with multiple queries

Combinatorial batch codes model the storage of a database on a given number of servers such that any k or fewer items can be retrieved by reading at most t items from each server. A combinatorial batch code with parameters n; k; m; t can be represented by a system F of n (not necessarily distinct) sets over an m-element underlying set X, such that for any k or fewer members of F there exists a system of representatives in which each element of X occurs with multiplicity at most t. The main purpose is to determine the minimum N(n; k; m; t) of total data storage ∑FєF |F| over all combinatorial batch codes F with given parameters. Previous papers concentrated on the case t = 1. Here we obtain the first nontrivial results on combinatorial batch codes with t > 1. We determine N(n; k; m; t) for all cases with k ≤ 3t, and also for all cases where n ≥ ti m dk=tei2¢. Our results can be considered equivalently as minimum total size ∑FєF |F| over all set systems F of given order m and size n, which satisfy a relaxed version of Hall's Condition; that is, |UF’| ≥ |F’|/t holds for every subsystem F’ ⊆ F of size at most k.