A Bound for Finding Defective Samples in Threshold Group Testing

In this paper, we consider a threshold group testing (TGT) problem in which there are two thresholds to determine the output result, i.e., positive or negative. We aim to find a lower bound for decoding defective samples out of a set of a large population. To this end, we use the Fano's inequality theorem in the information theory. We show that for highly successful decoding of smaller defective samples, a group matrix should be designed to be dense. In addition, we conclude that as a gap between two thresholds grows, a group matrix is required to be denser to find defective samples with only a small number of tests.