Symmetric Subdesigns of Symmetric Designs

A symmetric design \((U, \mathcal{A})\) is a strong subdesign of a symmetric design \((V, \mathcal{B})\) if \(U \subseteq V\) and \(\mathcal{A}\) is the set of non-empty
intersections \(B \cap U\), where \(B \in \mathcal{B}\). We demonstrate three constructions of symmetric designs, where this notion is useful, and produce two new infinite families of symmetric designs with parameters \(v = \left(\frac{73^{m+1} – 64}{9}\right), k = 73^m,\lambda = 9 \cdot 73^{m-1}\) and \(v = 1+2(q + 1)\left(\frac{(q + 1)^{2m} – 1}{q+2}\right), k = (q + 1)^{2m}, \lambda = \frac{(q + 1)^{2m-1} (q + 2)}{2}\) where \(m\) is a positive integer and \(q = 2^p – 1\) is a Mersenne prime. The main tools in these constructions are generalized Hadamard matrices and balanced generalized weighing matrices.