Probabilistic Group Testing with a Linear Number of Tests

In probabilistic nonadaptive group testing (PGT), we aim to characterize the number of pooled tests necessary to identify a random k-sparse vector of defectives with high probability. Recent work has shown that
$n$
tests are necessary when
$k$
=
$w$
(
$n$
/ log n). It is also known that O (k log n) tests are necessary and sufficient in other regimes. This leaves open the important sparsity regime where the probability of a defective item is~ 1/ log
$n$
(or
$k$
= 8 (
$n$
/ log n)) where the number of tests required is linear in n. In this work we aim to exactly characterize the number of tests in this sparsity regime. In particular, we seek to determine the number of defectives λ (
$a$
)
$n$
/ log
$n$
that can be identified if the number of tests is an. In the process, we give upper and lower bounds on the exact point at which individual testing becomes suboptimal, and the use of a carefully constructed pooled test design is beneficial.