Algebraic Decoding Techniques by Using Grobner Bases

9314347 Reed这是一个研究项目，包括四个相互关联的主题。(1)译码技术和Grobner基：多项式观点；(2)译码到循环码的真最小距离；(3)译码代数几何码；(4)简化译码算法。解码问题将以理想的方式表述，并寻求与错误定位多项式和Grobner基的关系。初步结果表明，在某些特殊情况下，利用Grobner基构造代数译码方法可以译码到真最小距离以内的所有循环码。这些技术有望成为解码代数几何代码的有效手段。这里需要进行复杂性分析。总体目标是提出一种低复杂度的解码算法，该算法可以校正到真正的最小距离。＊＊＊

9314347 Reed This is a program of research including four inter-related topics. (1) decoding techniques and Grobner bases: a polynomial point of view (2) decoding cyclic codes up to their true minimum distance, (3) decoding algebraic geometry codes, and (4) simplified decoding algorithms. The decoding problem will be formulated in terms of ideals, and relationships with the error locator polynomial and Grobner bases will be sought. Preliminary results show that is is possible to construct algebraic decoding methods by the use of Grobner bases to decode all cyclic codes up to the true minimum distance in some special cases. These techniques promise to be useful as a means of decoding algebraic geometry codes. Much effort in complexity analysis will be needed here. The overall goal is to come up with a low complexity decoding algorithm that corrects up to the true minimum distance. ***