Design

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

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

We typically let $$b$$ denote the number of blocks. Further, it is clear that $$k = v_1$$.