Group testing (GT) is the Boolean version of spare signal recovery and, due to its simplicity, a marketplace for ideas that can be brought to bear upon related problems, such as heavy hitters, compressed sensing, and multiple access channels. The definition of a "good" GT varies from one buyer to another, but it generally includes (i) usage of nonadaptive tests, (ii) limiting to $O(k \log n)$ tests, (iii) resiliency to test noise, (iv) $O(k \mathrm{poly}(\log n))$ decoding time, and (v) lack of mistakes. In this paper, we propose $Gacha~GT$. Gacha is an elementary and self-contained, versatile and unified scheme that, for the first time, satisfies all criteria for a fairly large region of parameters, namely when $\log k < \log(n)^{1-1/O(1)}$. Outside this parameter region, Gacha can be specialized to outperform the state-of-the-art partial-recovery GTs, exact-recovery GTs, and worst-case GTs. The new idea Gacha brings to the market is a redesigned Reed--Solomon code for probabilistic list-decoding at diminishing code rates over reasonably-large alphabets. Normally, list-decoding a vanilla Reed--Solomon code is equivalent to the nontrivial task of identifying the subsets of points that fit low-degree polynomials. In this paper, we explicitly tell the decoder which points belong to the same polynomial, thus reducing the complexity and enabling the improvement on GT.

翻译：群体检测（GT）是稀疏信号恢复的布尔版本，因其简洁性而成为解决重击者、压缩感知、多址接入等相关问题的思想集散地。“优秀”GT的定义因需求而异，但通常包含：（i）使用非自适应测试，（ii）限制在$O(k \log n)$次测试内，（iii）对测试噪声具有鲁棒性，（iv）$O(k \mathrm{poly}(\log n))$解码时间，以及（v）零错误。本文提出**Gacha GT**——一个基础且自包含、通用且统一的方案，首次在参数范围较大时（即$\log k < \log(n)^{1-1/O(1)}$）满足所有标准。在此参数范围外，Gacha可通过特化超越现有最先进的部分恢复GT、精确恢复GT和最坏情况GT。Gacha引入的新颖思想是重新设计的里德-所罗门码，用于在较大字母表上实现降速率下的概率列表解码。通常，标准里德-所罗门码的列表解码等价于识别匹配低次多项式的点集这一非平凡任务。本文通过明确告知解码器哪些点属于同一多项式，降低了复杂度，从而实现了GT性能的提升。