Some New Bush-Type Hadamard Matrices of Order $100$ and Infinite Classes of Symmetric Designs

Abstract

There are at least 52432 symmetric $(100,45,20)$ designs on which ${\text{Frob}}_{10}\times {\mathbb{Z}}_2$ acts as an automorphism group. All these designs correspond to Bush-type Hadamard matrices of order 100, and each leads to an infinite class of twin designs with parameters
$$v=100({81}^m+{81}^{m-1}+\dots +81+1),k=45(81{)}^m,\lambda =20(81{)}^m,$$
and an infinite class of Siamese twin designs with parameters
$$v=100({121}^m+{121}^{m-1}+\dots +121+1),k=55(121{)}^m,\lambda =30(121{)}^m,$$
where $m$ is an arbitrary positive integer. One of the constructed designs is isomorphic to that used by Z. Janko, H. Kharaghani, and V. D. Tonchev [4].