Disjunct matrices, also known as cover-free families and superimposed codes, are combinatorial arrays widely used in group testing. Among their variants, those that satisfy an additional combinatorial property called inclusiveness form a special class suitable for computationally efficient and highly error-tolerant group testing under the general inhibitor complex model, a broad framework that subsumes practical settings such as DNA screening. Despite this relevance, the asymptotic behavior of the inclusive variant of disjunct matrices has remained largely unexplored. In particular, it was not previously known whether this variant can achieve an asymptotically positive rate, a requirement for scalable group testing designs. In this work, we establish the first nontrivial asymptotic lower bound on the maximum achievable rate of the inclusive variant, which matches the strongest known upper bound up to a logarithmic factor. Our proof is based on the probabilistic method and yields a simple and efficient randomized construction. Furthermore, we derandomize this construction to obtain a deterministic polynomial-time construction. These results clarify the asymptotic potential of robust and scalable group testing under the general inhibitor complex model.

翻译：非分离矩阵，亦称覆盖自由族与叠加码，是广泛应用于群测试的组合阵列。在其诸多变体中，满足额外组合性质（称为包含性）的矩阵构成了一类特殊的矩阵，适用于一般抑制复合模型下计算高效且高容错的群测试。该模型是一个宽泛的框架，涵盖了诸如DNA筛选等实际场景。尽管具有这种相关性，非分离矩阵包含变体的渐近行为在很大程度上仍未得到探索。特别是，此前尚不清楚该变体是否能实现渐近正速率，这是可扩展群测试设计的一个必要条件。在本工作中，我们首次建立了关于该包含变体最大可实现速率的非平凡渐近下界，该下界与已知的最强上界在相差一个对数因子的范围内相匹配。我们的证明基于概率方法，并产生了一个简单高效的随机化构造。此外，我们对该构造进行去随机化，得到了一个确定性的多项式时间构造。这些结果阐明了在一般抑制复合模型下实现鲁棒且可扩展群测试的渐近潜力。