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

代数函数域构造的纠错码的最小距离

基本信息

批准号：
14540127
负责人：
UEHARA Tsuyoshi
金额：
$ 2.24万
依托单位：
Saga University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540127/
关键词：
Algebraic Function Fields Algebraic Geometry Codes Minimum Distance of Codes Fundamental Units Teichmueller Groupoids Monodromy Representation Chow Varieties Minimal Free Resolution 整数環の巾底グラースマン多様体 1次自由分解 Hasseの単数指数 Stanley-Reisne

项目摘要

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

我们进行了代数几何代码的研究，即由代数函数域构造的纠错代码，以及代数数论、算术代数几何、代数几何和代数组合方面的相关研究。该项目的目的是构造代数几何代码。显式代数函数域并应用于确定它们的最小距离，这是估计其纠正错误能力的重要数字。代数几何代码，我们确定了某类代数几何代码C的最小距离d(C)，称为单点代数几何代码，详细地说，我们在第一学年证明了最小距离d(C)。如果C'满足某些条件，则单点代数几何代码C的值等于其Feng-Rao下界d'（C）。在第二学年，我们构造了除以下之外的代数几何代码。并计算了它们的Feng-Rao下界，结果发现一些代数几何代码的Feng-Rao下界大于相应的单点型代码。在代数数论中，我们研究了有理数上较低可拓度的代数数域的类数和单位群的结构，特别是库默尔可拓的四次数域，我们还关注了整数是否存在的问题。 8次阿贝尔域的环具有幂基。作为算术代数几何的研究，我们构造了算术几何范畴内的Teichmueller群群，并描述了Galois作用和单向表示（与共形场论相关）此外，我们还证明了博戈莫洛夫猜想，即如果阿贝尔簇中的不可约曲线不是，则同构于椭圆曲线，则其代数点对于 Neron-Tate 高度离散均匀分布。作为代数几何的研究，我们考虑了在 n 阶射影空间中估计 d 度 Chow 型石油循环度的问题，并研究了射影不变量的结果与一些希尔伯特多项式之间的联系。作为代数组合学的研究，我们研究了特别是，我们给出了具有较小线性分辨率的 Buchsbaum Stanley-Reisner 环的唯一非消失同调群的维数上限。