Hadamard matrices : construction by groups and examples by computer

阿达玛矩阵：通过群构建和计算机示例

• 批准号: 10640031
• 负责人: NIWASAKI Takashi
• 金额: $ 2.11万
• 依托单位: Ehime University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640031/
• 关键词: Hadamard matrix; finite groups; dihedral groups; Hadamard conjecture; 二面体群; コード; 2面体群; デザイン

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.

设 G 为 2n 阶二面体群，并设 A、B、C 和 D 为其子集。最近，H. Kimura介绍了一种在一定条件下为这些子集构造8n+4次Hadamard矩阵H(A,B,C,D)的方法。在本研究中，我们根据积分群环ZG 研究了这些子集的一些性质，并给出了一些小奇整数n 的例子，如下： 1．我们将此构造推广到任意 2n.2 阶群 G。我们给出了这种结构的一些变体。3．我们考虑了对 G（及其子集）的几个动作，这些动作保留了 A、B、C 和 D 的条件。G 的全态也被包括在内。4．我们研究了两种特殊情况：(1) A、B、C 和 D 是对称的；(2) 更强烈的是，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 中的四个平方和的问题。5。利用计算机，我们对除n = 15外的所有奇数n < 30的二面体群构造了Hadamard矩阵H(A,B,C,D)。看来来自二面体群的Hadamard矩阵并不在少数。此外，有趣的是，几乎所有示例都是 y 不变类型。

项目成果

H. Kimura and T. Niwasaki: "Some properties of Hadamard matrices coming from dihedral groups"Graphs and Combinatorics. (in printing).

H. Kimura 和 T. Niwasaki：“来自二面体群的 Hadamard 矩阵的一些性质”图和组合。

H.Kimura and T.Niwasaki: "Some properties of Hadamard matrices coming from dihedral groups"Graphs and Combinatorics. (発表予定).

H.Kimura 和 T.Niwasaki：“来自二面群的 Hadamard 矩阵的一些性质”图和组合学（即将呈现）。

H.Kimura and T.Niwasaki: "Some properties of Hadamard matrices coming from dihedral groups"Graphs and Combinatorics. (発表予定). -H.

H.Kimura 和 T.Niwasaki：“来自二面群的 Hadamard 矩阵的一些性质”图和组合学（即将呈现）。