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). It was extended to symmetric designs by (Chowla Ryser).
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.
References
- ↑ Browne, Malcolm W. (20 December 1988), "Is a Math Proof a Proof If No One Can Check It?", The New York Times, https://www.nytimes.com/1988/12/20/science/is-a-math-proof-a-proof-if-no-one-can-check-it.html?pagewanted=all
- 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
- "The Search for a Finite Projective Plane of Order 10", American Mathematical Monthly 98 (4): 305–318, 1991, doi:10.2307/2323798, http://www.cecm.sfu.ca/organics/papers/lam/
- van Lint, J.H., and R.M. Wilson (1992), A Course in Combinatorics. Cambridge, Eng.: Cambridge University Press.
External links
- Weisstein, Eric W.. "Bruck–Ryser–Chowla Theorem". http://mathworld.wolfram.com/Bruck-Ryser-ChowlaTheorem.html.
 |  |
