Bruck–Ryser–Chowla_theorem
Bruck–Ryser–Chowla theorem
Nonexistence result for combinatorial block designs
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).