  In group testing, simple binary-output tests are designed to identify a small
number $t$ of defective items that are present in a large population of $N$
items. Each test takes as input a group of items and produces a binary output
indicating whether the group is free of the defective items or contains one or
more of them. In this paper we study a relaxation of the combinatorial group
testing problem. A matrix is called $(t,\epsilon)$-disjunct if it gives rise to
a nonadaptive group testing scheme with the property of identifying a uniformly
random $t$-set of defective subjects out of a population of size $N$ with false
positive probability of an item at most $\epsilon$. We establish a new
connection between $(t,\epsilon)$-disjunct matrices and error correcting codes
based on the dual distance of the codes and derive estimates of the parameters
of codes that give rise to such schemes. Our methods rely on the moments of the
distance distribution of codes and inequalities for moments of sums of
independent random variables. We also provide a new connection between group
testing schemes and combinatorial designs.
