Strongly separable matrices for nonadaptive combinatorial group testing

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 d̄-separable matrices (d̄-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 d̄-SM, while a d̄-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 strongly d-separable matrix (d-SSM) for nonadaptive CGT, which is sandwiched between d-DM and d̄-SM. We show that a d-SSM has the identifying algorithm more efficient than a d̄-SM, as well as the largest rate no less than a d-DM. In addition, the general bounds on the largest rate of d-SSM are established. Moreover, by the random coding method with expurgation, we derive an improved lower bound on the largest rate of 2-SSM which is much higher than the best known result of 2-DM.