  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.
