Symmetric Subdesigns of Symmetric Designs

Abstract

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.