算术代数几何在经典码的构造及列表译码中的应用

The developments of modern information theory need more and more mathematics. As a new discipline, Arithmetic algebraic geometry studies algebraic curves over finite fields and the corresponding algebraic function fields from the viewpoint of algebraic number theory and algebraic geometry. There are many good results on arithmetic algebraic geometry, which have important applications in coding and cryptography theory. The project consists of two topics on coding theory as below: 1. The constructions of classic block error-correcting codes and the analysis of their parameters. More explicitly, we will generalize the techniques used for constructing linear codes over finite fields to construct linear codes over algebraic curves. We believe that it is effective to obtain optimal error-correcting codes based on the excellent performance of algebraic geometry codes. By using the structures and properties of the extension of algebraic function fields and the rational points of algebraic curves, we obtain a number of error-correcting codes with special structures. 2 The list decoding algorithm of error-correcting codes. In this project, we attempt to construct a number of error-correcting codes with efficient list decoding algorithms and optimal list decoding radius. Besides, we try to describe the list decoding algorithms for those error-correcting codes obtained previously, and analyze their list decoding radius. Finally, we study the explicit construction of subspace evasive set to decrease the list size of our list decoding algorithms.

现代信息论的发展需要越来越多的数学。算术代数几何是从代数数论和代数几何角度研究有限域上的代数曲线及其对应的代数函数域的一门新的学科，有很多很好的结果。在实践中，算术代数几何特别是有限域上代数曲线的算术理论在编码学和密码学中有重要的应用。本项目主要研究以下两个重要问题：1.经典分组纠错码的构造及相关界的分析。将线性码的构造推广到代数曲线的情形，鉴于代数几何码的优异特性以得到参数更优的纠错码，并利用代数曲线上的扩张和有理点的结构及性质给出一批具有特殊结构的纠错码的构造。2.纠错码的列表译码(List decoding)算法。利用代数函数域构作一批好的纠错码，给出有效的列表译码算法使其具有最优的列表译码半径，此外对于构作出的具有良好参数的纠错码，分析其列表译码半径，同时研究列表译码算法中的subspace-evasive集合的有效构造，以改进算法输出码字个数。

现代信息论的发展需要越来越多的数学。算术代数几何是从代数数论和代数几何角度研究有限域上的代数曲线及其对应的代数函数域的一门新的学科，有很多很好的结果。本项目是基于算术代数几何等数学工具研究编码理论中以下两个课题： 1、纠错码的列表译码。我们证明了随机码的删除列表译码半径可以达到Singleton界，且代数几何码具有好的删除列表性能；随机码的突发错误列表译码半径可以达到Singleton界，且循环码有最好的突发错误列表译码参数及算法；随机秩度量码的列表译码半径可以达到最好的Gilbert-Varshamov界。2、拟循环码的构造。利用代数函数域自同构作用在有理点上所得轨迹的性质，选取合适的椭圆曲线和除子，具体构造了一批拟循环近MDS的代数几何码。