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Explicit construction of algebraic geometry codes

代数几何代码的显式构造

基本信息

批准号：
18540038
负责人：
UEHARA Tsuyoshi
金额：
$ 2.21万
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：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540038/
关键词：
Error-correcting codes Algebraic geometry codes Algebraic cnnstruntion of codes Low density parity check codes Power bases of integer rings Siegel modular forms Stanley-Reisner rings 低密度パリティ検査 Siegel 保型形式 Stanley-Reisner 環エルミート符号最小

项目摘要

1. We have researched on Hermitian codes which are constructed by Hermitian curves, and have found a new method of constructing of them. By our method we have constructed new Hermitian codes different from known ones, that is, one of the one-point type, and we have further computed a lower bound for their minimum distances. As a result, we have proved that our Hermitian codes have better properties than ones of the one-point type.2. We carried out an investigation on finding an explicit linear basis of the trace-norm code, and have found its explicit form when the number of variables is less than or equal to 3.3. We have researched on algebraic construction of low density parity check codes, and have shown methods of constructions of them by vector spaces over a finite field and by non-abelian groups.4. We have studied on structure of integral bases of algebraic number fields, and have proved that there is only one case that the integer ring of a 2-elementary abelian field of degree greater than or equal to 8 is generated by a single integer.5. We have researched on Siegel modular forms, and have described the ring structure of Siegel modular forms of degree 2 over a ring containing 1/6.6. We have studied Stanley-Reisner rings which are locally complete intersections. As a result, we proved that locally complete intersection Stanley-Reisner rings are complete intersections if the corresponding simplicial complexes are of dimension more than 1 and are connected.

1.我们对由厄米特曲线构造的厄米特码进行了研究，找到了构造厄米特码的新方法。通过我们的方法，我们构造了与已知的不同的新埃尔米特码，即单点类型之一，并且我们进一步计算了它们的最小距离的下界。结果，我们证明了我们的厄米特码比单点型码具有更好的性能。2．我们对追踪范数码的显式线性基进行了研究，并发现当变量数量小于或等于 3.3 时其显式形式。研究了低密度奇偶校验码的代数构造，给出了用有限域上的向量空间和非阿贝尔群构造低密度奇偶校验码的方法。 4．研究了代数数域的整数基结构，证明了次数大于或等于8的二元阿贝尔域的整数环只有一种情况是由单个整数生成的。 5．我们研究了Siegel模形式，并描述了包含1/6.6的环上的2阶Siegel模形式的环结构。我们研究了 Stanley-Reisner 环，它是局部完全交集。结果，我们证明了如果相应的单纯复形的维数大于 1 并且是连通的，则局部完全交集 Stanley-Reisner 环是完全交集。