BIBDs

Balanced incomplete block designs (BIBDs) are a class of combinatorial designs consisting of a finite set V of v elements, called treatments, and a collection B of b blocks. Each block contains exactly k elements, and every element of V occurs in exactly r blocks. Moreover, every pair of distinct elements of V occurs together in exactly lambda blocks. These conditions imply the two standard relations vr = bk and r(k − 1) = lambda(v − 1).

A BIBD is called symmetric if b = v; in that case r = k and the incidence structure (the relation between elements and blocks) is represented by a square incidence matrix. Fisher's inequality states that for nontrivial designs (where v > k) one has b ≥ v, with equality if and only if the design is symmetric.

A common special case is lambda = 1, which yields a Steiner system S(2, k, v). The Fano plane is the classic symmetric BIBD with parameters v = 7, b = 7, r = 3, k = 3, lambda = 1; it serves as a fundamental example in finite geometry.

Applications of BIBDs include experimental design, where blocks help balance comparisons among treatments and reduce confounding when a complete design is impractical. They also have connections to coding theory and finite geometry, since the incidence matrices of BIBDs relate to linear codes and geometric configurations.