A Note on Siamese Twin Designs Intersecting in a BIBD and a PBD |
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Authors: | Dean Crnković Ronan Egan |
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Institution: | 1.Department of Mathematics,University of Rijeka,Rijeka,Croatia |
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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. |
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