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# Combinatorial batch codes and transversal matroids

• Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers each of which stores a subset of the items. It is required that, for prescribed integers $k$ and $t$, any $k$ items can be retrieved by reading at most $t$ items from each server. Only the case $t=1$ is considered here. An optimal combinatorial batch code is one in which the total storage required is a minimum. We establish an important connection between combinatorial batch codes and transversal matroids, and exploit this connection to characterize optimal combinatorial batch codes if $n=m+1$ and $n=m+2$.
Mathematics Subject Classification: Primary: 05B30, 05B35, 05D05, 68P20, 68R05.

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