An incidence system (,
, , , )
in which a set
of points is partitioned into a family of subsets (blocks) in such a way that any two points determine
blocks with points in each block, and each point is contained in different blocks. It is also generally
required that ,
which is where the "incomplete" comes from in the formal term most often
encountered for block designs, balanced incomplete block designs (BIBD).

The five parameters are not independent, but satisfy the two relations

(1)

(2)

A BIBD is therefore commonly written as simply (, ,
), since and are given in terms of , ,
and by

(3)

(4)

A BIBD is called symmetric if (or, equivalently, ).

Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary
Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson).
New York: Wiley, pp. 1-12, 1992.Ryser, H. J. "The -Configuration."
§8.1 in Combinatorial
Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 96-102, 1963.