量子码与自对偶MDS码的构造

量子码和自对偶最大距离可分(MDS)码在通信系统和信息安全等领域有着广泛运用。本项目主要利用代数、数论和组合等工具构造参数良好的量子码和新自对偶MDS码，具体研究内容如下：研究本原Bose-Chaudhuri-Hocquenghem(BCH)码，构造参数良好的量子码；利用非本原 BCH码，构造更多具有较好参数的量子码；研究广义Reed-Solomon码，构造新量子MDS码；利用广义Reed-Solomon码，给出自对偶MDS码的构造。本项目的研究成果不仅能丰富代数编码理论，而且可以直接为量子通信和量子计算的实现与应用提供关键技术支撑。

Quantum codes and self-dual maximum distance separable (MDS) codes are widely used in communication system, information security and so on. This project focuses on the construction of quantum codes with good parameters and new self-dual MDS codes by the tools such as algebra, number theory, combinatorics and so on. The details are shown as follows. We research primitive BCH codes, and construct quantum codes with good parameters. We construct quantum codes with good parameters by nonprimitive BCH codes. We research generalized Reed-Solomon codes and construct some new quantum MDS codes. We construct self-dual MDS codes through generalized Reed-Solomon codes. The results obtained will enrich the theory of algebraic coding, and provide important technical support for the implementation and application of the quantum communications and quantum computing.