Strongly separable matrices for nonadaptive combinatorial group testing

TL;DR

This paper introduces strongly $d$-separable matrices (d-SSM) for nonadaptive group testing, combining advantages of existing structures to achieve high testing efficiency with weaker requirements, and provides improved bounds for their rates.

Contribution

It proposes the novel concept of strongly $d$-separable matrices, which match the identifying power of $d$-disjunct matrices but with weaker conditions, and establishes bounds on their rates.

Findings

d-SSM has the same identifying ability as d-DM.

Established bounds on the rate of d-SSM.

Improved lower bound for 2-SSM rate using random coding.

Abstract

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.