有限域上BCH码的结构理论及其应用研究

有限域上的BCH码在数字通信系统和数据存储系统中已被广泛应用，其理论研究一直是纠错码理论研究的热点和重点。本项目主要研究BCH码及其对偶码理论，并将其应用于构造性能优的量子纠错码。首先，利用分圆陪集理论，确定非狭义BCH码的维数，构造维数最优的线性码。并利用有限域上的代数曲线和特征理论，确定BCH码的最小距离，构造最优线性码。其次，利用有限域上的代数曲线理论，确定BCH码的对偶码的最小距离，构造距离最优的线性码。最后，利用交换环理论，确定BCH码的Hull维数，通过延长和截短BCH码的方法，构造性能优的量子纠错码。本项目的研究旨在丰富BCH码理论，并且给出构造量子纠错码的新途径，为可靠的数字数据通信和量子通信提供理论依据。

BCH codes have been widely used in digital communication systems and data storage systems, and research on BCH codes has been always a hot topic. This project mainly studies the theory of BCH codes and their dual codes, and their applications to the construction of quantum error-correcting codes with optimal parameters. Firstly, by using the cyclotomic cosets, the dimension of non-narrow BCH codes is determined, and the linear codes with optimal dimension are constructed. The minimum distance of BCH codes is also determined by using the algebraic curve theory and the character theory in finite field, and the linear codes with optimal parameters are constructed. Secondly, the minimum distance of dual codes of BCH codes is studied by using the algebraic curve theory in finite field, and some linear codes with optimal distance are constructed. Finally, the hull dimension of BCH codes is determined via the theory of commutative rings. By using the method of extending and puncturing BCH codes, quantum codes with optimal parameters are constructed. The purpose of this project is to enrich the theory of BCH codes and present a new way to construct quantum error-correcting codes, which can provide theoretical basis for reliable digital data communication and quantum communication.