Almost separable matrices

An m×n matrix A with column supports {Si} is k-separable if the disjunctions ⋃i∈KSi are all distinct over all sets K of cardinality k. While a simple counting bound shows that m>klog2n/k rows are required for a separable matrix to exist, in fact it is necessary for m to be about a factor of k more than this. In this paper, we consider a weaker definition of ‘almost k-separability’, which requires that the disjunctions are ‘mostly distinct’. We show using a random construction that these matrices exist with m=O(klogn) rows, which is optimal for k=O(n1−β) . Further, by calculating explicit constants, we show how almost separable matrices give new bounds on the rate of nonadaptive group testing.