  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.
