Algebraic Decoding Techniques by Using Grobner Bases

小行星9314347 这是一个研究计划，包括四个相互关联的主题。 (1)解码技术和Grobner基地：一个多项式的观点（2）解码循环码的真正最小距离，（3）解码代数几何码，（4）简化解码算法。 解码问题将制定的理想，并与错误定位多项式和Grobner基地的关系将寻求。 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是可能的，它可以对所有循环码进行真正的最小距离译码。 这些技术有望成为有用的解码代数几何代码的一种手段。 这里需要在复杂性分析方面做大量工作。 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. ***