nLab block design

Given positive integers $v,\beta,r,k,\lambda$, a block 2-design of type $(v,\beta,r,k,\lambda)$ is a combinatorial structure consisting of

• a finite set $X$ whose elements are declared points
• set $B$ of $k$-element subsets of $X$, called blocks

such that

• (i) for each $x\in X$ the number of $v\in B$ such that $x\in b$ is independent of $x$ and equals $r$
• (ii) for each pair of distinct elements $x,y\in X$, the number of $b\in B$ such that $\{x,y\}\subset b$ is independent of $(x,y)$ and equals $\lambda$

Block $s$-designs of type $(v,\beta,r,k,\lambda)$ for $s\gt 2$ have (ii) changed to

(ii’) for each subset $\{x_1,\ldots,x_s\}\subset X$ of cardinality precisely $s$, the number of $b\in B$ such that $\{x_1,\ldots,x_s\}\subset b$ is independent of the choice of such a subset and equals $\lambda$.

Examples: Steiner systems, finite projective planes.

Applications: cryptography, codes, finite geometries, finite group theory.

• Wikipedia block design