Combinatorial batch codes

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. In this paper, 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.