本项目主要研究序列的相关值，循环码,线性码，量子码,和补对偶码（LCD码）。具体研究内容如下：1）利用深刻数论和指数和等工具，研究m序列和其采样的相关值的Helleseth猜想，改进指数和Weil界，确定冯-罗指数和的精确值，给出m序列和其采样的相关值分布，计算单项式和多项式的Walsh谱，构造最优和几乎最优的码本；2）利用指数和，特征标， p-adic理论，计算一些特殊指数和的值，确定两个零点循环码的重量分布，构造最优或者几乎最优循环码和线性码，计算Galois环和p-adic域上的不可约循环码相应的本原幂等元和它的最小距离；3）利用广义Reed-Solomon码和矩阵方法，给出新的MDS量子码和MDS补对偶码。

This program mainly investigates correlation values of sequences, cyclic codes, linear codes, quantum codes and linear codes with complement dual (LCD codes). The details are given as follows: 1）by deep number theory and exponential sum and so on, investigating Helleseth’s conjecture about cross correlation values of m sequences and their decimations，improving Weil bound of exponential sum, determining the precise values of Feng-Luo exponential sum, giving distributions of cross correlation values of sequences and their decimations, calculating distributions of Walsh transformation of sequences, constructing optimal or almost optimal codebooks; 2) by exponential sum, characteristic matrix and p-adic theory, computing values of some special exponential sums, determining weight distributions of cyclic codes with two zeros, constructing optimal or almost optimal cyclic codes and linear codes, computing primitive idempotent elements associated with irreducible cyclic codes over Galois ring or p-adic field, and their minimum distances; 3) by generalized Reed-Solomon code and matrix theory, obtaining new MDS quantum codes and MDS LCD codes.