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Fast GMD Decoding of Codes from Algebraic Curves

代数曲线代码的快速 GMD 解码

基本信息

批准号：
10650354
负责人：
SAKATA Shojiro
金额：
$ 1.28万
依托单位：
The University of Electro-Communications
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10650354/
关键词：
algebraic geometric codes codes from curves soft-decision decoding GMD (generalized minimum distance) decoding erasure-and-error decoding BMS (Berlekamp-Massey-Sakata) algorithm erasure-addition algorithm erasire-deletion algorithm RS符号

项目摘要

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.

本研究的目的是扩展我们以前的一点代数几何代码（代码从代数曲线或曲面）的快速解码方法，这些代码的快速GMD（广义最小距离）解码。对于包括RS码在内的传统代数码的快速GMD译码，其他研究者已经给出了几种替代方法。另一方面，基于代数几何码是传统代数码从一维到n维的自然扩展这一认识，我们在《Grobner Bases and Applications》和《Codes，Curves and Signals》以及《Journal of IEICE》和《Mathematical Science》杂志上发表了一些综述性的论文。首先，在这个广泛的角度来看，我们发表了一篇论文的另一个版本的快速GMD解码的一维代数码在IEICE交易。接下来，我们与美国和丹麦的研究人员在IEEE Transactions on Information Theory上联合发表了一篇关于一点代数几何码的快速纠删译码方法的论文，该论文的内容应该是基于BMS算法的一点代数几何码快速GMD译码方法的核心。但是，仍然存在一个困难的问题，我们如何可以免除这些擦除和错误解码过程中所需的多数逻辑在确定未知的校正子值所需的错误校正到设计的距离的许多冗余的迭代。为了解决这个问题，我们提出了一对GMD过程，即擦除添加和擦除删除，可以在快速GMD解码过程中以多种替代方式相互结合。目前，我们正在尝试构造一种启发式算法，用于快速GMD译码方法。结合这些研究工作，我们在代数几何码的快速译码及其并行实现等方面发表了多篇相关论文。