Group Divisible Covering Designs with Block Size 4: A Type of Covering Array with Row Limit


  • Contract grant sponsor: NSERC; Contract grant number: OGP0170220.


A k-GDCD, group divisible covering design, of type inline image is a triple inline image, where V is a set of gu elements, inline image is a partition of V into u sets of size g, called groups, and inline image is a collection of k-subsets of V, called blocks, such that every pair of elements in V is either contained in a unique group or there is at least one block containing it, but not both. This family of combinatorial objects is equivalent to a special case of the graph covering problem and a generalization of covering arrays, which we call CARLs. In this paper, we show that there exists an integer inline image such that for any positive integers g and inline image, there exists a 4-GDCD of type inline image which in the worst case exceeds the Schönheim lower bound by δ blocks, except maybe when (1) inline image and inline image, or (2) inline image, inline image, and inline image or inline image. To show this, we develop constructions of 4-GDCDs, which depend on two types of ingredients: essential, which are used multiple times, and auxiliary, which are used only once in the construction. If the essential ingredients meet the lower bound, the products of the construction differ from the lower bound by as many blocks as the optimal size of the auxiliary ingredient differs from the lower bound.