Nonadaptive group testing with random set of defectives

In a group testing scheme, a set of tests is designed to identify a small number $t$ of defective items that are present among a large number $N$ of items. Each test takes as input a group of items and produces a binary output indicating whether any defective item is present in the group. In a non-adaptive scheme designing a testing scheme is equivalent to the construction of a disjunct matrix, an $M \times N$ binary matrix where the union of supports of any $t$ columns does not contain the support of any other column. In this paper we consider the scenario where defective items are random and follow simple probability distributions. In particular we consider the cases where 1) each item can be defective independently with probability $\frac{t}{N}$ and 2) each $t$-set of items can be defective with uniform probability. In both cases our aim is to design a testing matrix that successfully identifies the set of defectives with high probability. Both of these models have been studied in the literature before and it is known that $O(t\log N)$ tests are necessary as well as sufficient (via random coding) in both cases. Our main focus is explicit deterministic construction of the test matrices amenable to above scenarios. One of the most popular ways of constructing test matrices relies on \emph{constant-weight error-correcting codes} and their minimum distance. We go beyond the minimum distance analysis and connect the average distance of a constant weight code to the parameters of the resulting test matrix. With our relaxed requirements, we show that using explicit constant-weight codes (e.g., based on algebraic geometry codes) we may achieve a number of tests equal to $O(t \frac{\log^2 N}{ \log t})$ for both the first and the second cases.