Hadamard matrices : construction by groups and examples by computer

Research Project

Project/Area Number	10640031
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Algebra
Research Institution	Ehime University
Principal Investigator	NIWASAKI Takashi (1999) Ehime University, Mathematical Science, Assistant, 理学部, 助手 (50218252) 木村浩 (1998) 愛媛大学, 理学部, 教授 (70023570)
Co-Investigator(Kenkyū-buntansha)	SASAKI Hiroki Ehime University, Mathematical Science, Assoc. Prof., 理学部, 助教授 (60142684) KISO Kazuhiro Ehime University, Mathematical Science, Assistant, 理学部, 教授 (60116928) NOGURA Tsugunori Ehime University, Mathematical Science, Professor, 理学部, 教授 (00036419) SHAKHMATOV Dmitri Ehime University, Mathematical Science, Assoc. Prof., 理学部, 助教授 (90253294) HIRAIDE Koichi Ehime University, Mathematical Science, Assoc. Prof., 理学部, 助教授 (50181136) 宮本雅彦筑波大学, 数学系, 教授 (30125356) 大森博之愛媛大学, 教育学部, 教授 (20036370) 庭崎隆愛媛大学, 理学部, 助手 (50218252)
Project Period (FY)	1998 – 1999
Project Status	Completed (Fiscal Year 1999)
Budget Amount *help	¥3,300,000 (Direct Cost: ¥3,300,000) Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000) Fiscal Year 1998: ¥2,200,000 (Direct Cost: ¥2,200,000)
Keywords	Hadamard matrix / finite groups / dihedral groups / Hadamard conjecture / 二面体群 / コード / 2面体群 / デザイン
Research Abstract	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.