A Note on Siamese Twin Designs Intersecting in a BIBD and a PBD

Abstract

Suppose there exists a Hadamard 2-$(m,\frac{m-1}{2},\frac{m-3}{4})$ design with skew incidence matrix, and a conference graph with v vertices, where $v = 2m-1$. Under this assumption we prove that there exists a Siamese twin Menon design with parameters $(4m^{2},2m^{2}-m,m^{2}-m)$ intersecting in a balanced incomplete block design $\mathrm {BIBD}(2m^{2} - m, m^{2} - m, m^{2} - m - 1)$ and a pairwise balanced design $\mathrm {PBD}(2m^{2} - m, \{m^{2}, m^{2} - m\}, m^{2} - m - 1)$. These Menon designs lead to regular amicable Hadamard matrices of orders not previously constructed. Further we construct complex orthogonal designs of order $4m^2$ and Butson Hadamard matrices $\mathrm {BH}(4m^{2},2k)$ for all k. Some results regarding automorphisms of the constructed Menon designs are proven.