Symmetric BIBDs
The case of equality in Fisher's inequality, that is, a 2-design with an equal number of points and blocks, is called a symmetric design. Symmetric designs have the smallest number of blocks amongst all the 2-designs with the same number of points.
In a symmetric design r = k holds as well as b = v, and, while it is generally not true in arbitrary 2-designs, in a symmetric design every two distinct blocks meet in λ points. A theorem of Ryser provides the converse. If X is a v-element set, and B is a v-element set of k-element subsets (the "blocks"), such that any two distinct blocks have exactly λ points in common, then (X, B) is a symmetric block design.
The parameters of a symmetric design satisfy
This imposes strong restrictions on v, so the number of points is far from arbitrary. The Bruck–Ryser–Chowla theorem gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of these parameters.
The following are important examples of symmetric 2-designs:
Read more about this topic: Block Design