Definition:Balanced Incomplete Block Design

Definition
A Balanced Incomplete Block Design or BIBD with parameters $v, b, r, k, \lambda$ is a block design where:


 * $v$ is the number of points in the design
 * $b$ is the number of blocks
 * $k$ is the size of each block
 * $r$ is the number of blocks any point can be in
 * $\lambda$ is the number of times any two points can occur in the same block

and has the following properties:
 * Each block is of size $k$
 * All of the $\displaystyle \binom v 2$ pairs occur together in exactly $\lambda$ blocks.

A BIBD with parameters $v, b, r, k, \lambda$ is commonly written several ways, for example:
 * $\operatorname{BIBD} \left({v, k, \lambda}\right)$
 * $\left ({v, k, \lambda}\right)$-$\operatorname{BIBD}$

Properties
For any $\operatorname{BIBD} \left({v, k, \lambda}\right)$ the following are true:


 * $b k = r v$
 * $\lambda (v-1) = r (k-1)$
 * $\displaystyle \left({v, k, \lambda}\right)b = \frac{\binom v 2}{\binom k 2} \lambda = \frac{v(v-1)\lambda} {k(k-1)}$
 * $k < v$
 * $r > \lambda$

Note: All of the above are integers.

See Necessary Condition For The Existence of a BIBD for proofs of the above.

Also see

 * Fisher's Inequality: $b \ge v$.
 * Pairwise Balanced Design (PBD)