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