Algebraic investigation on Hadamard matrices, block designs and error-correcting codes

Hadamard 矩阵、分块设计和纠错码的代数研究

• 批准号: 62540073
• 负责人: ITO Noboru
• 金额: $ 1.02万
• 依托单位: Konan University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1987
• 资助国家: 日本
• 起止时间: 1987 至 1988
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-62540073/
• 关键词: Hadamard Tournament; Doubly Regular Asymmetric Digraph; Automorphism Group; Difference Set; Regular Hadamard Matrix; ハミング重み; アダマール行列; ブロックデザイン; 誤り訂正符号; 有向グラフ

Ito continues to work for doubly regular asymmetric digraphs(add. DRAD). He has shown that a DRAD with rank 4 automorphism groupsis the DRAD coming from (16,6,2) kummer design, and that DRADs with rank 5 automorphism groups such that one point stabilizers have two pairs of paired orbits are the DRADs coming from difference sets of quartic residues of DF(p) where p satisfies specific donditions. Ito has introduced the concept of N-graphs for DRADs, and with it characterized DRADs, together with their spectra, coming from regular Hadamard matrices. Ito is now investigating the existence problem (strong Hadamard conjecture) of Hadamard tournament (add. HT) of order v = 4m + 3. He remarks that we have only to consider the case where m is even. Then he has found an intersting connection between HT and a specific conjugacy class C of elements of order 3 in the group of orthogonal matrices of degree v over GF(2):There exists an element in C with certain conditions if and only if there exists a HT of order v. Ito is right now occupied with this investigation on HT. Ito is receiveing stimulating advices on computer programming for HT from Taguti and on geometric structure of DRADs from Hojo.

Ito 继续研究双正则不对称有向图（add. DRAD）。他证明了具有 4 阶自同构群的 DRAD 是来自 (16,6,2) kummer 设计的 DRAD，而具有 5 阶自同构群（使得一个点稳定器具有两对配对轨道）的 DRAD 是来自 DF(p) 的四次留数的不同集合的 DRAD，其中 p 满足特定条件。 Ito 引入了 DRAD 的 N 图概念，并用它描述了来自常规 Hadamard 矩阵的 DRAD 及其光谱。 Ito 现在正在研究阶 v = 4m + 3 的 Hadamard 锦标赛（add. HT）的存在问题（强 Hadamard 猜想）。他指出我们只需考虑 m 为偶数的情况。然后他发现了 HT 与 GF(2) 上 v 阶正交矩阵群中 3 阶元素的特定共轭类 C 之间的有趣联系：当且仅当存在 v 阶 HT 时，在某些条件下，C 中存在一个元素。伊藤现在正忙于对 HT 的研究。 Ito 正在从 Taguti 处获得关于 HT 计算机编程的建议，从 Hojo 处获得关于 DRAD 几何结构的建议。

项目成果

N.Ito.: Journal of Algebra. (1990)

N.Ito.：代数杂志。

M.Taguti.: International J.Computer Aided VLSI Design.

M.Taguti.：国际 J.计算机辅助 VLSI 设计。

N.Itoh: Discrete Methematics. 72. 181-185 (1988)

N.Itoh：离散数学。

S.Hojo: Proc.5th Sem on Finslar and Laprange Spaces Brasov,1987. (1988)

S.Hojo：Proc.5th Sem on Finslar 和 Laprange Spaces Brasov，1987。

N.Ito: "Doubly regular asymmetric digraphs" Discrete Mathematics. 72. 181-185 (1988)

N.Ito：“双重正则不对称有向图”离散数学。