Since there is some variation between authors of the terminology employed in design theory, we begin with some definitions. An incidence structure is a triple D = (P, B, I), where:
An incidence structure D is said to be uniform with blocksize k if D has at least one block and all blocks contain exactly k points. A uniform incidence structure is called trivial if each k-subset of the point set appears as a block (at least once).
Let t ≥0 be an integer. Then an incidence structure D is said to be t--balanced if there exists an integer λ ≥1 such that each t--subset of the point set is contained in exactly λ blocks of D.
A near--linear space is an incidence structure in which every block contains at least two points and any two points lie in at most one block. A linear space is a near--linear space in which any two points lie in exactly one block. It is usual, when discussing near--linear spaces, to use the term line in place of the term block.
Let v, k, t and λ be integers with v ≥k ≥t ≥0 and λ ≥1. A t--design with v points and blocksize k is an incidence structure D = (P, B, I) where:
The category names for the different families of incidence structures are as follows: