Hadamard matrices : construction by groups and examples by computer

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Let G be a dihedral group of order 2n and let A,B, C and D be its subsets. Recently, H. Kimura introduced a method to construct a Hadamard matrix H (A, B, C, D) of degree 8n + 4 under certain conditions for these subsets. In this research, we studied some properties of these subsets in terms of the integral group ring ZG, and gave some examples for small odd integers n as follows :1. We generalized this construction to arbitrary groups G of order 2n.2. We gave some variants of this construction.3. We considered several actions on G (and on its subsets) that preserve the conditions for A, B, C and D. The holomorph of G was included.4. We studied two particular cases :(1) A, B, C and D are symmetric ;(2) more strongly, for a dihedral group G, they are y-invariant.In these cases, the conditions for A, B, C and D simplify to problems similar to that of four square sums in the group ring ZG.5. By using computer, we constructed Hadamard matrices H (A, B, C, D) from dihedral groups for all odd integers n < 30 except for n = 15.It seems that Hadamard matrices coming from dihedral groups are not a few. Furthermore it is interesting to note that almost all examples are y-invariant type.