blockdesign

Blockdesign is a term used in combinatorial design theory and statistics to describe a structured way of organizing a finite set of elements into blocks that meet predefined balance properties. The basic objects are a set V of v points (treatments) and a collection B of b blocks, where each block contains exactly k points. In a balanced incomplete block design (BIBD), every point occurs in exactly r blocks, and every pair of distinct points occurs together in exactly λ blocks. These parameters satisfy the standard relations vr = bk and r(k−1) = λ(v−1). When these equations hold and blocks are all the same size, the design is balanced; if, in addition, v = b and r = k, it is called symmetric.

Steiner systems are a prominent family of block designs: t-designs in which every t-subset of points is contained in exactly one block. The most studied are Steiner systems S(t, k, v); for t = 2 these include Steiner triple systems S(2,3,v), which exist for v ≡ 1 or 3 mod 6 and yield BIBDs with λ = 1. A classic example is the Fano plane, a symmetric (7,7,3,3,1) design where seven points and seven blocks of size three are arranged so every pair of points lies in exactly one block.

Other concepts include resolvable designs, where blocks can be partitioned into parallel classes that cover V, and cyclic designs, which have rotational symmetries. Block designs have applications in the design of experiments (to control nuisance variation), agriculture, clinical trials, and computer experiments, and intersect with coding theory and finite geometry. Historical development spans statistics, led by R. A. Fisher, and combinatorics, with contributions from Steiner, Kirkman, Bose, and others.