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

本研究的目的是开发高效的代数几何码译码算法和广义代数几何码译码算法，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，使这些译码方法能够在不久的将来在高度信息化的世界中实际应用于数字通信系统。作为我们的研究成果，我们明确了这些码的快速解码方法之间的几种关系。特别地，我们证明了用于解码的并行Berlekamp-Massey （BM）算法和欧几里得算法是相同的，并且多序列BM算法可以被一系列单序列BM算法所取代。此外，我们还证明了Welch-Belekamp算法可以作为BM算法的向量版本来实现。我们分别在2004年10月10日至13日在意大利帕尔马举行的2004年信息理论与应用国际研讨会（ISITA-2004）、2005年9月4日至9日在澳大利亚阿德莱德举行的IEEE 2005年信息理论国际研讨会（ISITA-2006）和2006年10月29日至11月1日在韩国首尔举行的2006年信息理论与应用国际研讨会（ISITA-2006）上发表了这些结果。与此研究相关，我们于2004年出版了欧洲数学学会j.j usesen和t.h hoeholdt的《纠错码教程》的日文翻译的编码理论教材《纠错码导论》，并于2005年在IEEE信息理论汇刊上发表了题为《实现Berlekamp-Massey-Sakata算法在一类代数曲线上解码码的收缩阵列架构》的论文。2006年，我们举办了几次辅导课，全面介绍了我们在AG码快速解码方面的研究成果。

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.