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In nonadaptive group testing, the main research objective is to design anefficient algorithm to identify a set of up to $t$ positive elements among $n$samples with as few tests as possible. Disjunct matrices and separable matricesare two classical combinatorial structures while one provides a more efficientdecoding algorithm and the other needs fewer tests, i.e., larger rate.Recently, a notion of strongly separable matrix has been introduced, which hasthe same identifying ability as a disjunct matrix, but has larger rate. In thispaper, we use a modified probabilistic method to improve the lower bounds forthe rate of strongly separable matrices. Using this method, we also improve thelower bounds for some well-known combinatorial structures, including locallythin set families and cancellative set families.

Improved Lower Bounds for Strongly Separable Matrices and Related Combinatorial Structures