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

伊藤继续研究双正则非对称有向图（add. DRAD）。他证明了秩为4的自同构群的DRAD是来自（16，6，2）库默设计的DRAD，秩为5的自同构群使得一点稳定子具有两对成对轨道的DRAD是来自DF（p）的四次剩余差集的DRAD，其中p满足特定条件.伊藤提出了DRAD的N-图的概念，并用它刻画了DRAD和它们的谱都来自正则阿达玛矩阵。伊藤现在正在调查的存在性问题（强阿达玛猜想）的阿达玛比赛（添加。HT）的顺序v = 4 m + 3。他说，我们只需考虑m是偶数的情况。然后，他发现了HT与GF（2）上v次正交矩阵群中3阶元素的特定共轭类C之间的一个有趣联系：C中存在满足一定条件的元素当且仅当存在v阶HT。伊藤正在接受来自塔古蒂的HT计算机编程和来自北条的DRAD几何结构的启发性建议。

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.