Fast GMD Decoding of Codes from Algebraic Curves

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The objective of this research is to extend our previous method for fast decoding of one-point algebraic geometric codes (codes from algebraic curves or surfaces) to a fast GMD (generalized minimum distance) decoding of these codes. For fast GMD decoding of conventional algebraic codes including RS codes, several alternative methods have been given by other researchers. On the other hand, based on the recognition that algebraic geometric codes are a natural extention of conventional algebraic codes from one dimension to n dimension, we published some survey papers in volumes Grobner Bases and Applications and Codes, Curves and Signals as well as in journals Journal of IEICE and Mathematical Science. First, in this broad perspective, we published a paper on another version of fast GMD decoding of one-dimensional algebraic codes in IEICE Transactions. Next, we published a paper on fast erasure-and-error decoding method of one-point algebraic geometric codes jointly with American and Danish researchers in IEEE Transactions on Information Theory, the contents of which should be a core of fast GMD decoding method of these codes based on BMS algorithm. But, there still remains a difficult problem of how we can dispense with many redundant iterations of these erasure-and-error decoding procedures which are required by majority logic in determining the unknown syndrome values necessary for error correction up to the designed distance. To settle this problem, we have proposed a pair of GMD procedures, i.e. erasure-addition and erasure-deletion which can be combined with each other in many alternative ways during a fast GMD decoding process. At present we are trying to construct a kind of heuristic algorithm for fast GMD decoding method. Together with these research works we have published several relevant papers on fast decoding of algebraic geometric codes and its parallel implementation, etc.