纠错编码理论的发展为实现快捷、完整的现代通信提供理论基础，利用纠错码理论处理密码学中的问题是当前研究热点之一。本课题研究下列内容：(1) R为有限域或有限环、G为亚循环群，研究群代数R[G]的左理想所确定的R上特殊线性码类的结构性质和优化码的构造等问题；(2) 研究整数剩余类环和多项式剩余类环上常循环码和准扭转码的结构分类和参数估计等问题；(3) 发展两类有限环上的指数和理论，研究这两类环上特定线性码类的重量分布和完全重量分布，并应用于构造秘密共享方案和认证码等密码学领域。运用有限环上的模理论、斜多项式理论和数论方法，实现研究内容在理论和方法上的创新，获得新的优化码类及相关结果具有重要的理论和应用价值。

The development of the theory of error-correcting codes is the theoretical basis for modern communication to implement the fast and complete transmission of information, and it is also a research hotspot to solve problems in cryptography by use of the theory of error-correcting codes. The project is organized as follows. (1) Let R be a finite field or a finite commutative ring and G be a metacyclic group. We study the structural properties and the construction of optimal codes for the special class of linear codes over R corresponding to left ideals of the group ring R[G]. (2) We investigate the structural properties, classification and the estimation of code parameters of constacyclic codes and quasi-twisted codes over residue class rings of integer rings and residue class rings of polynomial rings over finite fields, respectively. (3) We develop a theory for exponential sums over two special class of finite rings, calculate the weight distribution and the complete weight distribution of certain linear codes over these rings, and consider some applications in the construction of secret sharing schemes and authentication codes. Using module theory and skew polynomial theory over finite commutative rings and some useful methods in number theory, we expect to improve the theory and methods for linear codes provided with certain algebraic structures. New optimal codes and relevant results obtained are of important theoretical and application value.