Symmetric design

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In combinatorial mathematics, a symmetric design is a block design with equal numbers of points and blocks. Thus, it has the fewest possible blocks given the number of points (by Fisher's inequality).

That is, a symmetric design is a (v,b,r,k,λ)-design with b = v and r = k. Either of the latter two equations implies the other. In fact, the parameters of a symmetric design satisfy

λ(v - 1) = k (k - 1).

Clearly, this imposes strong restrictions on v, so the number of points is far from arbitrary. The Bruck-Ryser-Chowla Theorem gives necessary but not sufficient conditions.

A theorem of Ryser gives a different combinatorial condition for an incidence structure to be symmetric. If X is a v-element set (the "point set"), and B is a v-element class of k-element subsets (called "blocks"), and any two blocks have exactly λ points in common, then (X, B) is a symmetric design.