New combinatorial designs and their applications to group testing (Q1061433)

Let \(B_ 1,B_ 2,...,B_ b\) be non-empty proper subsets of a set S of \(\nu\) symbols. The design D is defined to be the collection of such subsets along with the set of symbols S. In this paper a new class of combinatorial designs, named as t complete designs and denoting this property by C(t), are introduced. Definition. A design D is said to be with property C(t) if for every t elements \(\theta_ 1,\theta_ 2,...,\theta_ t\in S:\) \(\cup_{j\in T}B_ j=S-\{\theta_ 1,\theta_ 2,...,\theta_ t\}\) where \(T=\{j| \theta_ 1\not\in B_ j\quad with\quad i=1,2,3,...,t\}.\) Some results on designs with property C(t) are given. The authors also discuss the applications of such designs to group testing experiments. In a group testing context designs with \(b<\nu\) are needed, and such designs with property C(1) are shown to exist for all \(\nu\geq 6\).