有限域上的代数曲线在纠错码构造中的几点应用

项目名称： 有限域上的代数曲线在纠错码构造中的几点应用

项目编号： No.11501493

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 马立明

作者单位： 扬州大学

项目金额： 18万元

中文摘要： 有限域上的代数曲线可以用代数数论和代数几何的工具来研究，一直是理论数学研究的方向。但是自从Goppa发现利用代数曲线构造出代数几何码，有限域上的代数曲线被越来越多的应用到编码和密码理论中，得到了许多深刻的结果。本项目是利用有限域上的代数曲线来构造纠错码的，我们主要研究以下问题。一是考虑Hermitian函数域或者其他含多有理点的代数函数域来得到含多有理点的函数域；二是利用类域论来构造参数性质好的码；最后是利用代数曲线理论设计出新的构造码的方法。通过利用代数曲线的理论来研究纠错码的构造，我们希望构造更多性能良好的码以及发现两者之间越来越多的关联。

中文关键词： 纠错码；代数曲线；代数函数域；代数几何码；有限域

英文摘要： Algebraic curves over finite fields can be studied from the viewpoints of both algebraic number theory and algebraic geometry, and have been the main interests of the pure mathematics. But after the invention of the algebraic geometry codes given by Goppa, algebraic curves over finite fields have been greatly applied to coding theory and cryptography theory, and many impressive results have been gained. This project is based on the constructions of error-correcting codes from algebraic curves over finite fields, and we mainly focus on the following research problems. First, we try to obtain function fields with many rational places by considering the subfields of Hermitian function fields or other function fields with many rational places; second, we try to construct good codes from class field theory; in the end, we try to provide some new methods to construct good codes from algebraic curves. By employing the theory of algebraic curves over finite fields to construct error-correcting codes, we hope that we can construct more new codes with good parameters and discover more and more inter-connections between algebraic curves and error-correcting codes.

英文关键词： Error-correcting codes;algebraic curves;algebraic function fields;algebraic geometry codes;finite fields