# 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 Existence of BIBD for proofs of the above.