On Almost Disjunct Matrices for Group Testing

Abstract

In a group testing scheme, a set of tests is designed to identify a small number t of defective items among a large set (of size N) of items. In the non-adaptive scenario the set of tests has to be designed in one-shot. In this setting, designing a testing scheme is equivalent to the construction of a disjunct matrix, an M ×N matrix where the union of supports of any t columns does not contain the support of any other column. In principle, one wants to have such a matrix with minimum possible number M of rows (tests). One of the main ways of constructing disjunct matrices relies on constant weight error-correcting codes and their minimum distance. In this paper, we consider a relaxed definition of a disjunct matrix known as almost disjunct matrix. This concept is also studied under the name of weakly separated design in the literature. The relaxed definition allows one to come up with group testing schemes where a close-to-one fraction of all possible sets of defective items are identifiable. Our main contribution is twofold. First, we go beyond the minimum distance analysis and connect the average distance of a constant weight code to the parameters of an almost disjunct matrix constructed from it. Next we show as a consequence an explicit construction of almost disjunct matrices based on our average distance analysis. The parameters of our construction can be varied to cover a large range of relations for t and N. As an example of parameters, consider any absolute constant ε > 0 and t proportional to N ^δ, δ > 0. With our method it is possible to explicitly construct a group testing scheme that identifies (1 − ε) proportion of all possible defective sets of size t using only $O\Big(t^{3/2}\sqrt{ \log(N/\epsilon)}\Big)$ tests (as opposed to O(t ²logN) required to identify all defective sets).