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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 TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionEhime 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)
KeywordsHadamard 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.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (3 results)

All Other

All Publications (3 results)

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

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Kimura and T. Niwasaki: "Some properties of Hadamard matrices coming from dihedral groups"Graphs and Combinatorics. (in printing).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Kimura and T.Niwasaki: "Some properties of Hadamard matrices coming from dihedral groups"Graphs and Combinatorics. (発表予定).

    • Related Report
      1999 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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