It states that if a (v, b, r, k, λ)-design exists with v = b (a symmetric block design), then: The theorem was proved in the case of projective planes by Bruck & Ryser (1949).
In the special case of a symmetric design with λ = 1, that is, a projective plane, the theorem (which in this case is referred to as the Bruck–Ryser 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 = q2 + q + 1, r = k = q + 1, λ = 1.
Since a projective plane of order 10 has been shown not to exist using a combination of coding theory and large-scale computer search,[1] the condition of the theorem is evidently not sufficient for the existence of a design.
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.