Group testing (GT) is the Boolean counterpart of compressed sensing and the marketplace of new ideas for related problems such as cognitive radio and heavy hitter. A GT scheme is considered good if it is nonadaptive, uses $O(k \log n)$ tests, resists noise, can be decoded in $O(k \operatorname{poly}(\log n))$ time, and makes nearly no mistakes. In this paper, we propose "Gacha GT", an elementary, self-contained, and unified randomized 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 that runs through this paper, using an analogy, is to ask every person to break her $9$-digit "phone number" into three $3$-digit numbers $x$, $y$, and $z$ and write $(b, x)$, $(b, y)$, and $(b, z)$ on three pieces of sticky notes, where $b$ is her "birthday". This way, one can sort the sticky notes by birthday to reassemble the phone numbers. This birthday--number code and other coded constructions can be stacked like a multipartite graph pyramid. Gacha's encoder will synthesize the test results from the bottom up; and Gacha's decoder will reassemble the phone numbers from the top down.

翻译：群组测试（GT）是压缩感知的布尔域对应方法，也是认知无线电、重击者等相关问题新思想的试验场。若一种GT方案具备非自适应、使用$O(k \log n)$次测试、抗噪声、解码复杂度为$O(k \operatorname{poly}(\log n))$且几乎无错误，则被视为优质方案。本文提出"Gacha GT"——一种基础、自包含且统一的随机方案，首次在较大参数区域（即$\log k < \log(n)^{1-1/O(1)}$时）满足所有准则。在该参数区域外，Gacha可通过特化设计超越现有最先进的局部恢复GT、精确恢复GT及最坏情形GT。贯穿全文的新思想可类比为：要求每个人将她的9位"电话号码"拆解为三个3位数$x$、$y$、$z$，并在三张便签上分别写下$(b, x)$、$(b, y)$、$(b, z)$，其中$b$是她的"生日"。如此，便可通过生日排序便签重组电话号码。这种生日-数字编码及其他构造性编码可像多部图金字塔般堆叠。Gacha编码器自底向上综合测试结果，解码器则自顶向下重组电话号码。