New bounds on the number of tests for disjunct matrices

Given $n$ items with at most $d$ of which being positive, instead of testing these items individually, the theory of combinatorial group testing aims to identify all positive items using as few tests as possible. This paper is devoted to a fundamental and thirty-year-old problem in the nonadaptive group testing theory. A binary matrix is called $d$-disjunct if the boolean sum of arbitrary $d$ columns does not contain another column not in this collection. Let $T(d)$ denote the minimal $t$ such that there exists a $t\times n$ $d$-disjunct matrix with $n>t$. $T(d)$ can also be viewed as the minimal $t$ such that there exists a nonadaptive group testing scheme which is better than the trivial one that tests each item individually. It was known that $T(d)\ge\binom{d+2}{2}$ and was conjectured that $T(d)\ge(d+1)^2$. In this paper we narrow the gap by proving $T(d)/d^2\ge(15+\sqrt{33})/24$, a quantity in [6/7,7/8].