Relaxations of Hall’s Condition: Optimal batch codes with multiple queries

Abstract

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 ?F?F |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 ? t? m dk=te?2?. Our results can be considered equivalently as minimum total size ?F?F |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.