Explicit construction of algebraic geometry codes

1. 研究了由厄米曲线构造的厄米码，找到了一种新的构造厄米码的方法。用我们的方法构造了不同于已知厄密码的新厄密码，即一点型的厄密码，并进一步计算了它们的最小距离的下界。结果证明了我们的厄密码比单点码具有更好的性质。我们对寻找跟踪范数代码的显式线性基进行了调查，并找到了当变量数小于或等于3.3时的显式形式。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它们的方法。我们研究了代数数域的积分基的结构，证明了大于等于8次的2-初等阿贝尔域的整数环是由单个整数生成的只有一种情况。研究了西格尔模形式，描述了含1/6.6环上2次西格尔模形式的环结构。我们研究了局部完全相交的Stanley-Reisner环。因此，我们证明了局部完全交Stanley-Reisner环是完全交环，如果对应的简单复形的维数大于1且连通。

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.