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。我们已经研究了由Hermitian Curves构建的Hermitian代码，并找到了一种新的构造方法。通过我们的方法，我们构建了与已知代码不同的新隐士代码，即单点类型之一，并且我们进一步计算了其最小距离的下限。结果，我们已经证明，我们的遗传码具有比单点类型的属性更好的属性。2。我们对找到痕迹代码的显式线性基础进行了研究，并在变量数量小于或等于3.3时找到了其明确形式。我们已经研究了低密度平价检查代码的代数构建，并显示了有限场和非亚伯利亚组的向量空间的构造方法。4。我们已经研究了代数数字字段的整体基础的结构，并证明只有一种情况是，一个大于或等于8的2-元素的ABELIAN度数的整数环由单个整数产生。5。我们已经研究了Siegel模块化形式，并在包含1/6.6的环上描述了Siegel模块化2度模块的环结构。我们已经研究了斯坦利 - 赖斯纳环，这是本地完整的交叉点。结果，我们证明，如果相应的简单复合物的尺寸超过1，并且连接在一起，则局部完整的stanley-reisner环是完整的交叉点。