  In recent years, the mathematical limits and algorithmic bounds for
probabilistic group testing having become increasingly well-understood, with
exact asymptotic thresholds now being known in general scaling regimes for the
noiseless setting. In the noisy setting where each test outcome is flipped with
constant probability, there have been similar developments, but the overall
understanding has lagged significantly behind the noiseless setting. In this
paper, we substantially narrow this gap by deriving exact asymptotic thresholds
for the noisy setting under two widely-studied random test designs: i.i.d.
Bernoulli and near-constant tests-per-item. These thresholds are established by
combining components of an existing information-theoretic threshold decoder
with a novel analysis of maximum-likelihood decoding (upper bounds), and
deriving a novel set of impossibility results by analyzing certain failure
events for optimal maximum-likelihood decoding (lower bounds). Our results show
that existing algorithmic upper bounds for the noisy setting are strictly
suboptimal, and leave open the interesting question of whether our thresholds
can be attained using computationally efficient algorithms.
