Minimum distance of error-correcting codes constructed by algebraic function fields

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We performed the. research of algebraic geometry codes which are error-correcting codes constructed from algebraic function fields, and related researches in algebraic number theory, arithmetic algebraic geometry, algebraic geometry and algebraic combinatrics. The aim of this project is to construct algebraic geometry codes explicitly applying algebraic function fields and to determine their minimum distances, which are.important numbers for estimating their abilities of correcting errors.In the research of algebraic geometry codes, we determined the minimum distance d(C) of certain type of algebraic geometry codes C, called one-point algebraic geometry codes, in the first academic year. Speaking in detail, we proved that the minimum distance d(C) of a one-point algebraic geometry code C is equal to its Feng-Rao lower bound d' (C) if C'satisfies some conditions. In the second, academic year, we construct algebraic geometry codes other than of one-point type, and computed their Feng -Ra … More o lower bounds. As a result, we found some algebraic geometry codes whose Feng-Rao lower bound are larger than the corresponding codes of-one-point type.As a research in algebraic number theory, we investigated the class number and the structure of the unit groups for algebraic number fields of lower extension degree over the rationals, specifically for quartic number fields of Kummer extension. Also we concerned ourselves with the question whether the integer ring of an abelian field of degree 8 hasa power basis.As a research in arithmetic algebraic geometry, we constructed the Teichmueller groupoids in the category of arithmetic geometry, and we described the Galois action and the monodromy representation (associated with conformal field theory) on the Teichmueller groupoids. Furthermore we proved the Bogomolov conjecture which states that if an irreducible curve in an abelian variety is not, isomorphic to an elliptic curve, then its algebraic points are distributed uniformly discretely for the Neron-Tate height.As a research in algebraic geometry, we considered the problem to estimate the degree of the Chow variety oil-cycles of degree d in the n-th projective space, and investigated a connection between resultants, which are projective invariants, and some Hilbert polynomials.As a reaearch in algebraic combinatrics, we investigated a minimal free resolution of the Stanley-Reisnerring of a simplicial complex. In particular, we give an upper bound on the dimension of the Unique non-vanishing homology group of a Buchsbaum Stanley-Reisner ring with linear resolution. Less