Design

Definition

Implicit definition

A design is a nondegenerate tactical configuration of type $(s,t)$ where $(s,t)\neq (1,1)$ and $s\leq t$. A design is also termed a BIBD or balanced incomplete block design.

The only possibilities for $(s,t)$ for a design are $s=1,t\geq 2$ and $s=t=2$.

A design of type $(1,t)$ is termed a $t$-design.

Designs are in contrast with geometric configurations, which are tactical configurations of type $(1,1)$.

Explicit definition

A $t-(v,k,v_{t})$ design is a set of $v$ points, together with a collection of $k$-subsets of points (called blocks), such that every $t$-set of points lies in exactly $v_{t}$ blocks. (here $t\geq 2$).

We typically let $b$ denote the number of blocks. Further, it is clear that $k=v_{1}$.