Synthetic study of algebraic combinatorics

该赠款的目标之一是通过对会议和讲习班的财政支持，广泛促进日本代数组合学的发展。在2004-2007财政年度，赠款部分用于支持第21、22、23、24届代数组合专题讨论会（新州大学、爱媛大学、东洋大学、爱媛大学）、RIMS年度专题讨论会、在九州大学举行的关于球体包装的两次COE研讨会以及代数组合小型专题讨论会（九州大学三次，小begakuin大学一次）等。我们还举办了4次日韩代数和组合学研讨会，为国际关系与合作作出了贡献。日本的代数组合学研究正在稳步发展。这一进展涵盖了不同的领域，如远正则图、关联方案、编码、设计、格、模形式。首席研究员最近的研究集中在欧几里数的研究上。与Etsuko Bannai合作，获得了定权紧4设计的分类、高斯紧4设计的分类以及两个同心球上最优紧4设计的分类。此外，我们与Suprijanto合作，成功地证明了我们可以从一些紧绷的欧几里得设计开始得到新的紧绷欧几里得设计。我们最近的工作包括两个同心球体上紧密欧几里得7-设计的分类。我们还证明了t$2s-3的任何s次对映t设计都具有q -多项式关联方案的结构，并从极大实数MUB中找到了t=5和s=4的新例子。我们开始研究相干构型是如何与紧密的欧几里得设计相联系的。我们成功地证明了H. Cohn （Bannai-Bannai-Bannai）意义上的两个与普遍最优码相关的关联方案的唯一性，并与Abdukhalikov和Suda共同证明了其中一个关联方案的高维类似物是由最大实数MUB得到的。少

One of the goal of this grant was to widely contribute to the advancement of algebraic combinatorics in Japan through the financial support for conferences and workshops. During the fiscal year 2004-2007, the grant was used to partly support The 21st, 22nd, 23rd, 24th Algebraic Combinatoric Symposiums(Shinshu University, Ehime University, Tobhiku University, Ehime University), annual symposiums at RIMS, two COE Workshops on Sphere Packings, held at Kyushu University, and Mini-symposiums on algebraic combinatorics(three at Kyushu university and one in Kobegakuin University), etc. We also held 4 Japan-Korea workshops on Algebra and Combinatorics, and contributed to the international relations and cooperations.Research in Algebraic Combinatorics in Japan is showing steady progress. The progress covers diverse areas such as distant-regular graphs, association schemes, codes, designs, lattices, modular forms. The recent research of the principal investigator is focused on the study of Eucli … More dean designs. Collaborating with Etsuko Bannai, we obtained the classification of tight 4-designs with constant weight, the classification of Gaussian tight 4-designs, and the classification of optimal tight 4-designs on two concentric spheres. In addition, jointly with Suprijanto, we succeeded in showing that we can get new tight Euclidean designs starting from some tight Euclidean designs. Our recent work includes the classification of tight Euclidean 7-designs on two concentric spheres. We have also proved that any antipodal t-design of degree s with t$2s-3 has the structure of Q-polynomial association scheme, and found such new examples with t=5 and s=4 from maximal real MUB. We started to study how coherent configurations are attached to tight Euclidean designs. We succeeded in proving the uniqueness of two association schemes related to universally optimal codes in the sense of H. Cohn (Bannai-Bannai-Bannai), and have shown, jointly with Abdukhalikov and Suda, that higher dimensional analogues of one of them are obtained from maximal real MUB. Less