New symmetric designs from regular Hadamard matrices

Abstract

For every positive integer $m$, we construct a symmetric $(v,k,\lambda )$-design with parameters $v={{h((2h-1)^{2m}-1)}\over{h-1}}$, $k=h(2h-1)^{2m-1}$, and $\lambda =h(h-1)(2h-1)^{2m-2}$, where $h=\pm 3\cdot 2^d$ and $|2h-1|$ is a prime power. For $m\geq 2$ and $d\geq 1$, these parameter values were previously undecided. The tools used in the construction are balanced generalized weighing matrices and regular Hadamard matrices of order $9\cdot 4^d$.