Fast decoding methods of algebraic geometry codes and generalized algebraic geometry codes

本研究的目的是开发不仅适用于代数几何（AG）码而且适用于广义代数几何（GAG）码的高效译码算法，并进一步将这些码的快速译码方法的过去和新的研究成果系统化，以便这些译码方法在不久的将来可以实际应用于高度信息技术世界的数字通信系统中。我们已经阐明了这些码的快速译码方法之间的几个关系。特别地，我们证明了并行Berlekamp-Massey（BM）译码算法和Euclidean译码算法是完全相同的，并且多序列BM译码算法可以用一系列单序列BM译码算法代替。此外，我们还证明了Welch-Belekamp算法可以实现为BM算法的矢量版本。我们在2004年信息理论与应用国际研讨会上介绍了这些结果（ISITA-2004），2004年10月10-13日在意大利帕尔马，2005年9月4-9日在澳大利亚阿德莱德举行的IEEE 2005信息理论国际研讨会，以及2006信息理论和应用国际研讨会（ISITA-2006），分别于2006年10月29日至11月1日在韩国首尔举行。与本研究相关，我们于2004年出版了一本关于编码理论的教科书，名为Introduction to Error-Correcting Codes，该教科书是J.Justesen和T.Hoeholdt的A Course in Error-Correcting Codes的日文翻译，由欧洲数学学会出版，以及题为“实现Berlekamp-Massey-Sakata算法的Syrup阵列架构，用于在一类代数曲线上解码代码”的论文IEEE Transactions on Information Theory，2005年。2006年，我们举办了几次辅导讲座，对我们在AG码快速译码方面的研究成果和最新成果进行了全面的回顾。

The objective of this research is to develop efficient algorithms for decoding not only algebraic geometric (AG) codes but also generalized algebraic geometric (GAG) codes and furthermore aim at systematizing past and new research results of fast decoding methods for those codes so that these decoding methods can be used practically in digital communication systems in the highly information-technological world in near future.As outcomes of our researches, we have clarified several relations among fast decoding methods of those codes. In particular, we have shown that the parallel Berlekamp-Massey (BM) algorithm and Euclidean algorithm for decoding are identical, and that the multiple-sequence BM algorithm can be replaced by a succession of single-sequence BM algorithm. Furthermore, we have shown that the Welch-Belekamp algorithim can be realized as a vectorial version of BM algorithm. We presented these results in the 2004 International Symposium on Information Theory and Applications (ISITA-2004), in Parma, Italy, October 10-13, 2004, in the IEEE 2005 International Symposium on Information Theory, in Adelaide, Australia, September 4-9, 2005, and in the 2006 International Symposium on Information Theory and Applications (ISITA-2006), in Seoul, Korea, October 29-November 1, 2006, respectively. Related to the research, we published a textbook on Coding Theory entitled Introduction to Error-Correcting Codes which is a Japanese translation of A Course in Error-Correcting Codes by J.Justesen, and T.Hoeholdt published by European Mathematical Society, in 2004, as well as a paper entitled "Systolic array architecture implementing Berlekamp-Massey-Sakata algorithm for decoding codes on a class of algebraic curves" in IEEE Transactions on Information Theory, in 2005. In 2006, we had several tutorial lectures to give a grand survey of past and new results of our researches on fast decoding of AG codes.