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.