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

我们执行了。代数几何代码的研究是由代数函数字段构建的错误校正代码以及代数数理论，算术代数几何，代数几何和代数几何和代数组合中的相关研究。该项目的目的是构建代数几何代码明确应用代数功能字段并确定其最小距离，这是估算其纠正错误能力的最重要数字。在代数几何法的研究中，我们确定了某些类型的Algebraic Gerement Gealtrysemess，以确定最低限度的距离d（c）。详细说明，我们证明了单点代数几何代码C的最小距离d（c）等于其风水下限d'（c），如果c'sating c'Asating c'Asating c'Asating c'Asating c'Asing c'Asing c'Asing c）。在第二学年中，我们构建了除一分点类型以外的代数几何代码，并计算了他们的冯-RA……更多的o下限。结果，我们发现了一些代数的几何代码，其风水下限大于一点点类型的相应代码。作为代数数理论的研究，我们研究了较低的延长程度的代数数字的班级编号和单位组的结构，用于Quartics Nubpersims neversemers extensims extermense of Kertension neversemers sextersims extermers。 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 provided 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 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结果是投影不变的，与一些希尔伯特多项式之间的联系。作为代数组合的重新搜索，我们研究了简单复合物的Stanley Reisnerring的最小自由分辨率。特别是，我们在Buchsbaum stanley-Reisner环的独特非变化同源群的维度上给出了上限，并具有线性分辨率。较少的