Symmetric design

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:$lambda\; (v-1)\; =\; k(k-1).$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.

References

* van Lint, J.H., and R.M. Wilson (1992), "A Course in Combinatorics". Cambridge, Eng.: Cambridge University Press.