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Our Laboratory conducts research on mathematical theory with algebraic or discrete approach. Our research topics are (1) algebraic combinatorics, (2) codes and lattices, (3) application of abstract algebra to computer science.
(1) The theory of association schemes has been developed to unify the application of linear programming to coding theory and design theory by Delsarte in 1970's. It generalizes the action of finite groups, and gives a framework for algebraic graph theory, algebraic coding theory and combinatorial design theory. In order to develop algebraic tools for these theories, we investigate applications of algebraic methods to combinatorics, mainly from graph spectra, finite groups, representation theory, linear algebra and optimization.
(2) A code is a subspace of a finite-dimensional vector space over a finite field. This seemingly simple concept has been widely used to study combinatorial problems using algebraic methods. Codes themselves can also be investigated from tools in number theory, modular forms in particular, via integral lattices. The class of self-dual codes is an interesting class of codes which give rise to unimodular lattices, and are related to sphere packing problem. We investigate the problems of construction and classification of codes and lattices.
(3) The theory of finite fields, elliptic curves, or more generally, algebraic curves and algebraic number theory, have been considered to belong to pure mathematics. These areas started to play an important role in coding theory and cryptography. We investigate applications of these areas to computer science from a theoretical viewpoint.
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