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

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

\[
v= 100(81^m + 81^{m-1} + \ldots + 81+1),\, k=45(81)^m ,\, \lambda=20(81)^m ,
\]

and an infinite class of Siamese twin designs with parameters

\[
v= 100(121^m + 121^{m-1} + \ldots + 121+1),\, k=55(121)^m ,\, \lambda=30(121)^m ,
\]

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].