Almost separable matrices

An m x n matrix A with column supports {S_i} is k-separable if the disjunctions U_iεkS_i 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(k log n) rows, which is optimal for k = O(n^1-β). Further, by calculating explicit constants, we show how almost separable matrices give new bounds on the rate of nonadaptive group testing.