Consecutive positive detectable matrices and group testing for consecutive positives

Colbourn (1999) developed some strategy for nonadaptive group testing when the items are linearly ordered and the positives items form a consecutive subset of all items. We improve his strategy by introducing the concept of 2-consecutive positive detectable matrices (2CPD-matrix) requiring that all columns and bitwise OR-sum of each two consecutive columns are pairwise distinct. Such a matrix is called maximal if it has a maximal possible number of columns with respect to some obvious constraints. Using a recursive construction we prove the existence of maximal 2CPD-matrices for any column size m ∈ N except for the case m = 3. Furthermore, we construct maximal 2CPD-matrices where each column is of some fixed constant weight. This leads to pooling designs, where each item appears in the same number of pools and all pools are of the same size.