On the Twin Designs with the Ionin-type Parameters

Let 4n2 be the order of a Bush-type Hadamard matrix with q = (2n − 1)2 a prime power. It is shown that there is a weighing matrix W (4(q + qm−1 + · · ·+ q + 1)n, 4qn) which includes two symmetric designs with the Ionin–type parameters ν = 4(q + qm−1 + · · ·+ q + 1)n, κ = q(2n − n), λ = q(n − n) for every positive integer m. Noting that Bush–type Hadamard matrices of order 16n2 exist for all n for which an Hadamard matrix of order 4n exist, this provides a new class of symmetric designs. ∗Thanks to Stephen Ney for proving me wrong on my first choice of the cyclic group by writing a program and applying the group. Wolfgang Holzmann, as always, was a great helper.