编码理论中几类Weil指数和及其应用

Weil指数和是一类特殊的指数和，其在编码理论中有着广泛的应用。本项目主要运用代数、数论、组合学等数学工具研究几类Weil指数和的值分布及其编码应用。具体如下: 1) 研究m-序列及其采样序列的互相关值分布，Helleseth猜想和其他公开问题；2) 给出广义Moisio型指数和的精确值分布，构造性能良好的线性码，计算单项式和多项式的Walsh谱；3) 有效改进一些特殊多项式指数和的Weil界。本项目研究将有利于编码理论研究方法和研究内容的丰富和发展，并且有望推进Weil指数和在有效纠错编码中的应用。

Weil exponential sums is a special kind of exponential sums, which is widely used in coding theory. This project mainly uses algebra, number theory, combinatorics and other mathematical tools to study the value distributions and coding applications of several types of Weil exponential sums. The details are as follows: 1) studying the distributions of cross-correlation values of m-sequences and their decimations, Helleseth's conjecture and other open problems; 2) giving the exact value distributions of generalized Moisio's exponential sums, constructing linear codes with good performance, and calculating the Walsh spectra of monomials and polynomials; 3) improving the Weil bounds of exponential sums of some special polynomials effectively. The research of this project will be conducive to the enrichment and development the research methods and research content in coding theory, and is expected to promote the application of Weil exponential sums in effective error-correcting coding.