Constructions of Optimal Sparse r -Disjunct Matrices via Packings

Abstract

Group testing has been widely used in various aspects, and the $r$-disjunct matrix plays a crucial role in group testing. The original purpose of the group testing is to identify a set of at most $r$ positive items from a batch of $M$ total items using as fewer tests as possible. In many practical applications, each test can include only a limited number of items and each item can participate in a limited number of tests. In this paper, we use the tools from combinatorial design theory to construct optimal 2-disjunct matrices with $n$ rows and limited row weight $3<\rho \le \lfloor \frac{n-1}{2}\rfloor$ and optimal 3-disjunct matrices with $n$ rows and limited row weight $4<\rho \le \lfloor \frac{n-1}{3}\rfloor$, respectively. Also we use the known graph-matching theorem to give an asymptotically optimal upper bound on the $r$-disjunct matrices with limited column weight $r+1\le w\le 2r$.