In recent years, the mathematical limits and algorithmic bounds for probabilistic group testing have 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).

翻译：近年来，概率群组检测的数学极限与算法边界已得到日益深入的理解，在无噪声场景的一般标度机制下现已建立精确的渐近阈值。在测试结果以恒定概率发生翻转的噪声场景中，虽已取得类似进展，但整体认知仍显著滞后于无噪声场景。本文通过推导两种广泛研究的随机测试设计——独立同分布伯努利测试与近似恒定每项测试次数——在噪声场景下的精确渐近阈值，大幅缩小了这一认知差距。这些阈值的建立融合了现有信息论阈值解码器的组件与最大似然解码的新颖分析（上界），同时通过分析最优最大似然解码的特定失效事件，推导出一系列新的不可能性结果（下界）。