用代数方法研究Galois自对偶码的构造和表示问题

自对偶码与数学中的其它领域如格理论、密码学、不变量理论、区组设计和图论等都有着密切联系，自对偶码的构造是纠错编码领域中广受关注的重要课题之一。参数最优或者优化自对偶码的构造尚无高效的系统构造方法，需要借助计算机搜索来实现。本项目探索运用代数学的深刻理论和丰富的方法，发展并完善相关理论，对Galois自对偶码（包含欧几里得和厄米特自对偶码为典型特例）展开深入研究，力争在以下两方面取得突破：（1）有限域和有限链环上具有特定阶数(型)置换自同构的Galois自对偶码的结构表示、参数性质和优化码的构造方法；（2）有限域和Galois环及其扩环（域）上的左群码、常循环码和加性常循环码中Galois自对偶码的深刻理论和优化码的有效构造方法。以期建立代数学理论与有限域（环）上Galois自对偶码的深刻联系，实现Galois自对偶码的构造和表示问题在理论和方法上的创新，提高构造效率，为计算机搜索指引方向。

Self-dual codes are closely related to other fields of mathematics, such as lattice theory, cryptography, invariant theory, block designs and graph theory, etc. The construction of self-dual codes is one of the most important topics in the theory of error-correcting codes. So far, There is no general and effective method to construct general self-dual codes with optimal or good parameters, and the construction usualy needs to be realized by computer searching. This project will explore the development and improvement of relevant theories by use of the profound theory and rich methods of algebra, and make an deep study on Galois self-dual codes (including Euclidean and Hermitian self-dual codes as typical special cases) , to strive for breakthroughs in the following two contents: (1) Study the algebraic structure, representation, parameter properties and the construction method of Galois self-dual codes with a permutation automorphism of certain special order (type) over finite fields and finite chain rings. (2) Develop the profound and detailed theory of Galois self-dual codes in left group codes, constacyclic codes and additive constacyclic codes over finite fields, Galois rings and their extended rings (fields), respectively; Get efficient and explicit construction methods for optimized Galois self-dual codes. Our project expects to establish a deep connection between algebra theory and Galois self-dual codes over finite fields (rings) , and realize the innovation in theory and method for the research of the construction and representation of Galois self-dual codes, to improve the efficiency of constructions and to direct the computer searching.