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A weighted module view of integral closures of affine domains of type I
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 
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 $c = 2, k2$ and $k1$. In addition, we obtain improved existence results for arbitrary fixed $c$ using the probabilistic method.
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