Rational congruence for uniform multiplicative designs
A multiplicative design is a family of n subsets of an n-set whose incidence matrix A satisfies \(A^ TA=D+\alpha \alpha^ T\) where \(\alpha\) is a positive real vector and D is a positive diagonal matrix. This is a generalization of \(\lambda\)-designs where \(\alpha\) is a constant vector and of (v,k,\(\lambda)\)-designs where D is additionally required to be scalar. In a uniform design we only require that D be scalar, so the equation is \(A^ TA=dI+\alpha \alpha^ T.\) One of the basic results on (v,k,\(\lambda)\)-designs is the Bruck-Ryser- Chowla Theorem which says that k-\(\lambda\) must be a square if v is even and that \(z^ 2=(k-\lambda)x^ 2+\lambda y^ 2(-1)^{(v-1)/2}\) must have a nontrivial integral solution if v is odd. This can be proved with or without reference to the theory of rational congruences. The purpose of this paper is to investigate the implications of the theory of rational congruences for the existence of uniform multiplicative designs. The Hass-Minkowski Theorem provides the main line of attack. The main result gives a finite set of equations, suitable for programming on a computer, which must be satisfied if there is a rational matrix satisfying the equation \(A^ TA=dI+\alpha \alpha^ T\) for a uniform design.
scientific article; zbMATH DE number 3987291