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August  2010, 4(3): 419-431. doi: 10.3934/amc.2010.4.419

Combinatorial batch codes and transversal matroids

Richard A. Brualdi 1, , Kathleen P. Kiernan 1, , Seth A. Meyer 1, and  Michael W. Schroeder 1,

1. 

Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States, United States, United States, United States

Received  January 2010 Revised  May 2010 Published  August 2010

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 $n=m+2$.
Keywords: presentation of a transversal matroid., transversal matroid, Combinatorial batch codes, cocircuit, dual matroid.
Mathematics Subject Classification: Primary: 05B30, 05B35, 05D05, 68P20, 68R0.
Citation: Richard A. Brualdi, Kathleen P. Kiernan, Seth A. Meyer, Michael W. Schroeder. Combinatorial batch codes and transversal matroids. Advances in Mathematics of Communications, 2010, 4 (3) : 419-431. doi: 10.3934/amc.2010.4.419
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