Explicit construction of algebraic geometry codes

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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.