# Definition:Balanced Incomplete Block Design

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## 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 Existence of BIBD for proofs of the above.

## Also see

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