# American Institute of Mathematical Sciences

February  2009, 3(1): 13-27. doi: 10.3934/amc.2009.3.13

## Combinatorial batch codes

 1 Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom 2 David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada 3 Department of Computer Science, Lakehead University, hunder Bay, ON, P7B 5E1, Canada

Received  July 2008 Revised  December 2008 Published  January 2009

In this paper, we study batch codes, which were introduced by Ishai, Kushilevitz, Ostrovsky and Sahai in [4]. A batch code specifies a method to distribute a database of $n$ items among $m$ devices (servers) in such a way that any $k$ items can be retrieved by reading at most $t$ items from each of the servers. It is of interest to devise batch codes that minimize the total storage, denoted by $N$, over all $m$ servers.
We restrict out attention to batch codes in which every server stores a subset of the items. This is purely a combinatorial problem, so we 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 present batch codes that are optimal with respect to the storage requirement, $N$. We also study uniform codes, where every item is stored in precisely $c$ of the $m$ servers (such a code is said to have rate $1/c$). Interesting new results are presented in the cases $c = 2, k-2$ and $k-1$. In addition, we obtain improved existence results for arbitrary fixed $c$ using the probabilistic method.
Citation: M. B. Paterson, D. R. Stinson, R. Wei. Combinatorial batch codes. Advances in Mathematics of Communications, 2009, 3 (1) : 13-27. doi: 10.3934/amc.2009.3.13
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