Research on Lee Error Correcting AG Codes

本研究主要是针对Lee距离优于Hamming距离的多值系统，研究其纠错码，特别是代数几何码。（1）从理论上推导了扩展广义Reed-Muller码的最小Lee距离和最小Hamming距离，并阐明了在许多参数下，最小Lee距离超过最小Hamming距离。(2)虽然人们认为代数几何码优于传统码，但人们澄清，当冗余符号的数量相对较少时，BCH码可以优于代数几何码。上级。(3)比较了Fermat曲线和Fermat曲面上的代数几何码，阐明了利用Fermat曲面不可能得到更好的代数几何码[1]。(4)在文献[2]中，给出了代数几何码子域子码维数的一个改进下界。(5)研究了有限域GF（p）上的BCH码与有限整数环Z_上的BCH码之间的关系<pk>[3]。参考文献[1]水谷次郎：“论构造在代数曲面上的代数几何码“，东京工业大学毕业论文，2月，1995. [2]龙太郎松本：“代数几何码的子字段子码维数的改进下界“，毕业论文，东京工业大学，2005年2月，1996. [3]松谷重则：“有限环Z_上的BCH码<pk>“，毕业论文，东京工业大学，2004年2月，1996.

This research was performed to investigate error correcting codes, especially algebraic geometric codes, for multi-valued systems where the Lee distance is preferred to the usual Hamming distance. The research results are summarized as follows :(1) The minimum Lee and Hamming distances of the extended generalized Reed-Muller codes were derived theoretically and it was clarified that in many parameters the minimum Lee distance exceeds the minimum Hamming distance.(2) Though it was thought that the algebraic geometric codes are superior to the conventional codes, it was clarified that when the number of redundant symbols is relatively small the BCH codes can be better than the algebraic geometric codes.(3) The algebraic geometric code on Fermat curve and on Fermat surface were compared and it was clarified that it is not possible to get better codes by using Fermat surface [1].(4) An improved lower bound for the dimension of subfield subcodes of algebraic geometric codes was derived [2].(5) The relationship between the BCH codes over the finite field GF (p) and the BCH codes over the finite integer ring Z_<pk> was investigated [3].References[1] Jiro Mizutani : "On the Algebraic Geometric Codes Constructed on Algebraic Surfaces, " Graduation Thesis, Tokyo Institute of Technology, Feb., 1995.[2] Ryutaroh Matsumoto : "Improved Lower Bound for the Dimension of Subfield Subcodes of Algebraic Geometric Codes, " Graduation Thesis, Tokyo Institute of Technology, Feb., 1996.[3] Shigenori Kasuya : "On the BCH Codes over the Finite Integer Ring Z_<pk>, " Graduation Thesis, Tokyo Institute of Technology, Feb., 1996.