Bruck–Ryser–Chowla theorem

The BruckRyserChowla theorem is a result on the combinatorics of block designs. It states that if a (v, b, r, k, λ)-design exists with v = b (a symmetric block design), then:

  • if v is even, then k λ is a square;
  • if v is odd, then the following Diophantine equation has a nontrivial solution:
    x2 (k λ)y2 (1)(v1)/2 λ z2 = 0.

The theorem was proved in the case of projective planes in (Bruck & Ryser 1949). It was extended to symmetric designs in (Ryser & Chowla 1950).

Projective planes

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. 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,[1] 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 (v, b, r, k, λ)-design is equivalent to the existence of a v × v incidence matrix R with elements 0 and 1 satisfying

R RT = (k λ)I + λJ

where I 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.


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