Improved bounds for group testing designs

Group testing designs (GTDs), both adaptive and nonadaptive, are useful in reducing the number of tests needed to identify the defective items from a given set of at least six items. In this paper, we obtain improved bounds on the number of group tests necessary for both adaptive and nonadaptive GTDs. It is established that any nonadaptive GTD needs at least 2n group tests for identifying all the defective items from a group of 2n items having at most 2 defective items. In the same context, an adaptive multistage GTD with a maximum of 2n group tests is presented here. It is further shown that under restrictions on group size, optimal nonadaptive GTDs can be constructed using Generalized Petersen Graphs. Also presented is the construction of a family of two-stage adaptive GTDs that are useful under certain conditions.