The BruckâÂÂRyserâÂÂChowla theorem is a result on the combinatorics of symmetric block designs that implies nonexistence of certain kinds of design. It states that if a -design exists with (implying and ), then:
The theorem was proved in the case of projective planes by . It was extended to symmetric designs by .
In the special case of a symmetric design with û = 1, that is, a projective plane, the theorem (which in this case is referred to as the BruckâÂÂRyser theorem) can be stated as follows: If a finite projective plane of order q exists and q is congruent to 1 or 2 (mod 4), then q must be the sum of two squares. Note that for a projective plane, the design parameters are v = b = q<sup>2</sup> + q + 1, r = k = q + 1, û = 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.
The existence of a symmetric (v, b, r, k, û)-design is equivalent to the existence of a v àv incidence matrix R with elements 0 and 1 satisfying
where is the v àv identity matrix and J is the v àv all-1 matrix. In essence, the BruckâÂÂRyserâÂÂChowla theorem is a statement of the necessary conditions for the existence of a rational v àv matrix R satisfying this equation. In fact, the conditions stated in the BruckâÂÂRyserâÂÂChowla theorem are not merely necessary, but also sufficient for the existence of such a rational matrix R. They can be derived from the HasseâÂÂMinkowski theorem on the rational equivalence of quadratic forms.