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 (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)^(v−1)/2 λ z² = 0.

The theorem was proved in the case of projective planes by Bruck & Ryser (1949). It was extended to symmetric designs by Chowla & Ryser (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 R T = (k - λ) I + λ J

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.

References

^ Browne, Malcolm W. (20 December 1988), "Is a Math Proof a Proof If No One Can Check It?", The New York Times

Bruck, R.H.; Ryser, H.J. (1949), "The nonexistence of certain finite projective planes", Canadian Journal of Mathematics, 1: 88–93, doi:10.4153/cjm-1949-009-2, S2CID 123440808
Chowla, S.; Ryser, H.J. (1950), "Combinatorial problems", Canadian Journal of Mathematics, 2: 93–99, doi:10.4153/cjm-1950-009-8, S2CID 247194753
Lam, C. W. H. (1991), "The Search for a Finite Projective Plane of Order 10", American Mathematical Monthly, 98 (4): 305–318, doi:10.2307/2323798, JSTOR 2323798
van Lint, J.H., and R.M. Wilson (1992), A Course in Combinatorics. Cambridge, Eng.: Cambridge University Press.