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:
- x2 − (k − λ)y2 − (−1)(v−1)/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, 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.
References
- Bruck, R.H.; Ryser, H.J. (1949), "The nonexistence of certain finite projective planes", Canadian J. Math., 1: 88–93, doi:10.4153/cjm-1949-009-2
- Chowla, S.; Ryser, H.J. (1950), "Combinatorial problems", Canadian J. Math., 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
- van Lint, J.H., and R.M. Wilson (1992), A Course in Combinatorics. Cambridge, Eng.: Cambridge University Press.