Research on Lee Error Correcting AG Codes

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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.