两类指数和的相关性质及应用

项目名称： 两类指数和的相关性质及应用

项目编号： No.11201275

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 张天平

作者单位： 陕西师范大学

项目金额： 22万元

中文摘要： 指数和研究中关于Kloosterman和及二项指数和性质的讨论在加法数论、解析数论、模形式、椭圆曲线以及编码研究中占有举足轻重的位置，并和很多数论难题如"Linnik猜想"、"Selberg特征值猜想"、 "Sato-Tate猜想"以及"华林问题"等密切相关。本项目主要针对带特征Kloosterman和及二项指数和的上界估计、高次均值、混合均值等性质进行深入研究，拟采用指数和及特征和估计方法，始终把握在指数和及特征和估计中"均值估计优于单个估计"的原则，并结合"分段"、"分类"、"凑项"、"整合"、"转换"等技巧，以期获得一些较强渐近公式，进一步揭示这两类指数和的均值分布规律，补充和拓展指数和的相关理论。作为应用，拟对Kloosterman 码的编码性质进行探索和推广，并基于高次Kloosterman 和及二项指数和尝试构造新的线性码，以期为编码理论的研究和发展提供新的研究内容与方法支持。

中文关键词： 指数和；Kloosterman 和；二项指数和；特征和；

英文摘要： The study on the properties of two kinds of exponential sums such as Kloosterman sums and the two-term exponential sums occupies a pivotal position in additive number theory, analytic number theory, modular forms, elliptic curves, and coding theory. And many famous number theoretic problems such as "Linnik conjecture", "Selberg's eigenvalue conjecture", "Sato-Tate conjecture" and "Waring's problem" are closely related. In the project the upper bound, high power mean and hybrid mean of the general Kloosterman sums and the two-term exponential sums with character are to be studied by using the methods of estimating the exponential sums and character sums. Many techniques are to be applied in order to obtain some sharp asymptotic formulae, through which the distributive properties of the two sums are revealed, and the theory involving them is complemented. As an application, the coding properties of Kloosterman code are to be explored and promoted. Furthermore, some new linear codes based on the high power Kloosterman and the two-term exponential sums are to be constructed, with an attempt to provide the coding theory new research contents and methods of development.

英文关键词： Exponential sums；Kloosterman sums；two-term exponential sums；character sums；