Abstract
In this paper, we study batch codes, which wereintroduced by Ishai, Kushilevitz, Ostrovsky and Sahai in [4].A batch code specifies a method to distribute adatabase of $n$ items among $m$ devices (servers)in such a way that any $k$ itemscan be retrieved by reading at most $t$ items from each of the servers. It is of interest to devise batch codes thatminimize the total storage, denoted by $N$, over all $m$ servers.We restrict out attention to batch codesin which every server stores a subset ofthe items. This is purely a combinatorial problem, sowe call this kind of batch code a ''combinatorial batch code''.We only study the special case $t=1$, where,for various parameter situations, we are able to presentbatch codes that are optimal with respect to the storagerequirement, $N$. We also study uniform codes, where every item isstored in precisely $c$ of the $m$ servers (such a codeis said to have rate $1/c$). Interesting new resultsare presented in the cases $c = 2, k-2$ and $k-1$. In addition,we obtain improved existence results for arbitraryfixed $c$ using the probabilistic method.
Highlights
Kushilevitz, Ostrovsky and Sahai [3] have shown that problems connected with reducing the computational overhead of private information retrieval can be related to the question of how to distribute a database of n items among m devices so that any k items can be retrieved by reading at most t items from each of the servers [3]
We have defined combinatorial batch codes to be set systems whose points represent the items in a database, with the servers being represented by subsets of these points
We say that an (n, N, k, m)-combinatorial batch code (CBC) is optimal if N ≤ N ′ for all (n, N ′, k, m)-CBC and we denote the corresponding value of N by N (n, k, m)
Summary
Kushilevitz, Ostrovsky and Sahai [3] have shown that problems connected with reducing the computational overhead of private information retrieval can be related to the question of how to distribute a database of n items among m devices (servers) so that any k items can be retrieved by reading at most t items from each of the servers [3]. In this paper we consider batch codes for which the decoding is reading; these are referred to as replication-based batch codes in [3] In this case, each server can be represented as a subset of the alphabet set, so the problem of constructing such codes falls naturally within a combinatorial framework. Each server can be represented as a subset of the alphabet set, so the problem of constructing such codes falls naturally within a combinatorial framework We call these codes “combinatorial batch codes” and define them as follows. We are only interested in (n, N, k, m) combinatorial batch codes with N < kn and N > n
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