# Bruck–Ryser–Chowla theorem

The **Bruck–Ryser–Chowla 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:*x*^{2 }− (*k*− λ)*y*^{2}− (−1)^{(v−1)/2}λ*z*^{2}= 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* = *q*^{2} + *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 - Chowla, S.; Ryser, H.J. (1950), "Combinatorial problems",
*Canadian Journal of Mathematics*,**2**: 93–99, doi:10.4153/cjm-1950-009-8 - 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.