In nonadaptive combinatorial group testing (CGT), it is desirable to identify a small set of up to $d$ defectives from a large population of $n$ items with as few tests (i.e. large rate) and efficient identifying algorithm as possible. In the literature, $d$-disjunct matrices ($d$-DM) and $\bar{d}$-separable matrices ($\bar{d}$-SM) are two classical combinatorial structures having been studied for several decades. It is well-known that a $d$-DM provides a more efficient identifying algorithm than a $\bar{d}$-SM, while a $\bar{d}$-SM could have a larger rate than a $d$-DM. In order to combine the advantages of these two structures, in this paper, we introduce a new notion of \emph{strongly $d$-separable matrix} ($d$-SSM) for nonadaptive CGT and show that a $d$-SSM has the same identifying ability as a $d$-DM, but much weaker requirements than a $d$-DM. Accordingly, the general bounds on the largest rate of a $d$-SSM are established. Moreover, by the random coding method with expurgation, we derive an improved lower bound on the largest rate of a $2$-SSM which is much higher than the best known result of a $2$-DM.

中文翻译：

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用于非自适应组合组测试的强可分离矩阵

在非自适应组合组测试 (CGT) 中，希望使用尽可能少的测试（即大比率）和有效的识别算法从大量的 $n$ 项目中识别出一小组最多 $d$ 的缺陷品。在文献中，$d$-disjunct 矩阵 ($d$-DM) 和 $\bar{d}$-separable 矩阵 ($\bar{d}$-SM) 是两个被研究了几十年的经典组合结构. 众所周知，$d$-DM 提供了比 $\bar{d}$-SM 更有效的识别算法，而 $\bar{d}$-SM 可能比 $d 具有更大的识别率$-DM。为了结合这两种结构的优点，在本文中，我们为非自适应 CGT 引入了 \emph{strongly $d$-separable matrix} ($d$-SSM) 的新概念，并证明了 $d$- SSM 具有与 $d$-DM 相同的识别能力，但要求比 $d$-DM 弱得多。因此，建立了 $d$-SSM 的最大利率的一般界限。此外，通过带有去除的随机编码方法，我们推导出了 $2$-SSM 的最大速率的改进下限，该下限远高于 $2$-DM 的最佳已知结果。