In this paper, we study batch codes, which were introduced by Ishai, Kushilevitz, Ostrovsky and Sahai in . 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.\ud 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
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