Synthetic study of algebraic combinatorics

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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