Abstract
Combinatorial batch codes were defined by Paterson, Stinson, andWei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers each of which stores a subset of the items. It is required that, for prescribed integers $k$ and $t$, any $k$ items can be retrieved by reading at most $t$ items from each server. Only the case $t=1$ is considered here. An optimal combinatorial batch code is one in which the total storage required is a minimum. We establish an important connection between combinatorial batch codes and transversal matroids, and exploit this connection to characterize optimal combinatorial batch codes if $n=m+1$ and $n=m+2$.
Highlights
In [7], batch codes are defined as a way to model the efficient retrieval of n items of information distributed among m servers where any string of k items is retrievable by reading at most t items from each of the servers
Our contribution in this paper is the following: we first show that in an optimal CBC(n, m, k), each server contains at least one item, that a matrix presentation A of an optimal CBC(n, k, m) has term rank ρ(A) equal to m. We develop this important connection between CBCs and transversal matroids
We review the necessary facts about transversal matroids and their presentations that we need for our discussion of optimal CBCs
Summary
In [7], batch codes are defined as a way to model the efficient retrieval of n items of information distributed among m servers where any string of k items is retrievable by reading at most t items from each of the servers. A CBC(n, N, k, m) corresponds to the tranversal matroid of a family B of m subsets of a set X of n elements in which every subset of X of cardinality k is independent; the size N is the sum of the cardinalities of the sets in B, equivalently, the number of 1s in its incidence matrix presentation. Our contribution in this paper is the following: we first show that in an optimal CBC(n, m, k), each server contains at least one item, that a matrix presentation A of an optimal CBC(n, k, m) has term rank ρ(A) equal to m We develop this important connection between CBCs and transversal matroids. Using the connection between CBCs and transversal matroids, we determine N (m + 2, k, m), a question that was left open in [9]
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