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.
Highlights
Batch codes were introduced by Ishai, Kushilevitz, Ostrovsky and Sahai [10]
They represent the distributed storage of an n-element database on a set of m servers when any k or fewer data items can be recovered by submitting a limited number t of queries to each server
This model can be used for amortizing the computational cost in
Summary
A combinatorial batch code with parameters n, k, m, t can be represented with its ‘dual’ set system (shortly, CBC∗(n, k, m, t)-system) F , where the m elements of the underlying set correspond to the m servers and the members of F correspond to the n items of data. The system containing the underlying set X as member with multiplicity n is a CBC∗(n, k, m, t) and N (n, k, m, t) is well-defined. Systems which are CBC∗(n, k, m, t) and have minimum total size N (n, k, m, t) will be called optimal.
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