THE BASIC STRUCTURE OF t - DESIGNS

One simile that solitary shines In the dry desert of a thousand lines. The collection of all subsets of cardinality k of a set with v elements (k < v) has the property that any subset of t elements, with 0 ≤ t ≤ k, is contained in precisely v−t k−t subsets of size k. The subsets of size k provide therefore a nice covering for the subsets of a lesser cardinality. Observe that the number of subsets of size k that contain a subset of size t depends only on v, k, and t and not on the specific subset of size t in question. This is the essential defining feature of the structures that we wish to study. The example we just described inspires general interest in producing similar coverings without using all the v k subsets of size k but rather as small a number of them as possi