Bruck–Ryser–Chowla theorem


The Bruck–Ryser–Chowla theorem is a result on the combinatorics of block designs that implies nonexistence of certain kinds of design. It states that if a -design exists with v = b, then:
The theorem was proved in the case of projective planes in. It was extended to symmetric designs in.

Projective planes

In the special case of a symmetric design with λ = 1, that is, a projective plane, the theorem can be stated as follows: If a finite projective plane of order q exists and q is congruent to 1 or 2, then q must be the sum of two squares. Note that for a projective plane, the design parameters are v = b = q2 + q + 1, r = k = q + 1, λ = 1. Thus, v is always odd in this case.
The theorem, for example, rules out the existence of projective planes of orders 6 and 14 but allows the existence of planes of orders 10 and 12. Since a projective plane of order 10 has been shown not to exist using a combination of coding theory and large-scale computer search, the condition of the theorem is evidently not sufficient for the existence of a design. However, no stronger general non-existence criterion is known.

Connection with incidence matrices

The existence of a symmetric -design is equivalent to the existence of a v × v incidence matrix R with elements 0 and 1 satisfying
where is the v × v identity matrix and J is the v × v all-1 matrix. In essence, the Bruck–Ryser–Chowla theorem is a statement of the necessary conditions for the existence of a rational v × v matrix R satisfying this equation. In fact, the conditions stated in the Bruck–Ryser–Chowla theorem are not merely necessary, but also sufficient for the existence of such a rational matrix R. They can be derived from the Hasse–Minkowski theorem on the rational equivalence of quadratic forms.