On the twin designs with the Ionin-type parameters (Q1964486)

A Bush-type Hadamard matrix is a block matrix of order \(4n^2\) with block size \(2n\) such that the diagonal blocks are matrices of all \(1\)'s and the off-diagonal blocks have row sum zero. If \(q= (2n- 1)^2\) is a prime power, it is shown that there is a weighing matrix with elements \(0\), \(\pm 1\) and orthogonal rows and columns which becomes a symmetric design with Ionin-type parameters by replacing either all \(1\)'s or all \(-1\)'s by zeros.