Hadamard matrices : construction by groups and examples by computer

设G是2n阶二面体群，A、B、C、D是G的子集。最近，H.Kimura提出了一种在一定条件下构造8n+4次Hadamard矩阵H(A，B，C，D)的方法。在本研究中，我们利用整数群环ZG研究了这些子集的一些性质，并给出了一些关于小奇数n的例子如下：1.我们把这种构造推广到2n2阶的任意群G。我们给出了这种结构的一些变体。我们考虑了在G(及其子集)上保持A、B、C和D条件的几种作用。我们研究了两种特殊情况：(1)A，B，C和D是对称的；(2)更强的是，对于二面体群G，它们是y不变的.在这些情况下，A，B，C和D的条件简化为类似于群环ZG.5中的四个平方和的问题.用计算机从二面体群构造了除n=15外的所有奇数n&lt；30的Hadamard矩阵H(A，B，C，D)。此外，有趣的是注意到几乎所有的例子都是y不变类型。

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.