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

我们表演了。代数几何码（由代数函数域构造的纠错码）的研究，以及代数数论、算术代数几何、代数几何和代数组合的相关研究。本课题的目的是明确地应用代数函数域构造代数几何码，并确定它们的最小距离，即。评估他们纠正错误能力的重要数字。在代数几何码的研究中，我们在第一学年确定了一类称为单点代数几何码的代数几何码C的最小距离d(C)。详细地证明了单点代数几何码C的最小距离d(C)等于它的Feng-Rao下界d‘ (C)，如果C’满足某些条件。在第二学年，我们构造了非单点型的代数几何码，并计算了它们的Feng -Ra - More下界。结果，我们找到了一些丰饶下界大于相应的一点型码的代数几何码。作为代数数论的一项研究，我们研究了有理数上的低扩展次代数数域，特别是Kummer扩展的四次数域的类数和单位群的结构。并讨论了8次阿贝尔域的整数环是否具有幂基的问题。作为算术代数几何的一项研究，我们构造了算术几何范畴内的Teichmueller群样，并描述了Teichmueller群样上的伽罗瓦作用和与共形场论相关的单态表示。进一步证明了Bogomolov猜想，该猜想指出，如果一个阿贝尔变量中的不可约曲线与椭圆曲线不同构，那么它的代数点对于Neron-Tate高度是均匀离散分布的。作为代数几何中的一项研究，我们考虑了在n次射影空间中估计d次的Chow变化油循环的阶数问题，并研究了射影不变量的结果与一些Hilbert多项式之间的联系。作为代数组合学中的一项研究，我们研究了简单复合体的Stanley-Reisnerring的最小自由分辨率。特别地，我们给出了具有线性分辨率的Buchsbaum - Stanley-Reisner环的唯一不消失同调群的维数上界。少

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