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

我们执行了。代数几何码的研究，即由代数函数域构造的纠错码，以及代数数论、算术代数几何、代数几何和代数组合学的相关研究。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，这是估计它们纠错能力的重要数值.在代数几何码的研究中，我们在第一学年确定了一类代数几何码C（称为单点代数几何码）的最小距离d（C）.具体地说，我们证明了：当C '满足一定条件时，一点代数几何码C的最小距离d（C）等于它的Feng-Rao下界d'（C）。在第二学年，我们构造了非一点型代数几何码，并计算了它们的Feng-Ra ...更多信息 o下限。作为代数数论的研究，我们研究了有理数域上低扩张度代数数域，特别是库默扩张的四次数域的类数和单位群的结构.作为算术代数几何的一个研究，我们构造了算术几何范畴内的Teichmueller群胚，并描述了Teichmueller群胚上的Galois作用和单值表示（与共形场论有关）.进一步证明了Bogomolov猜想，即如果交换簇中的不可约曲线不同构于椭圆曲线，则其代数点对于Neron-Tate高度是均匀离散分布的.作为代数几何的研究，我们考虑了n阶射影空间中d次Chow簇油圈的阶估计问题，并研究了结式之间的联系，作为代数组合学的一个补充，我们研究了单纯复形的Stanley-Reisnerring的一个极小自由分解。特别地，我们给出了具有线性分解的BuchsbaumStanley-Reisner环的唯一非零同调群的维数的一个上界。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