New Bounds on the Number of Tests for Disjunct Matrices

Abstract

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 × 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) ≥ ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d+2</sup> ) and was conjectured that T(d) ≥ (d + 1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . In this paper, we narrow the gap by proving T(d)/d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ≥ (15 + √33)/24, a quantity in [6/7,7/8].