编码理论是数学与计算机科学交叉学科，为可靠通信提供方法（纠错码）与数学基础。纠错码的一个重要任务是通过增加尽量少的冗余使得所传输消息具有尽量大的纠错能力，另一个重要任务就是如何对于接收向量进行快速有效地纠错。Reed-Solomon码因为其较强的纠错能力与快速的编译码算法等优点，是最早被应用在工程中的代数编码技术之一，被广泛的应用于各种商业用途，最显著的是在CD、DVD和蓝光光盘等上的使用。于是，Reed-Solomon码的译码算法的改进与译码性能界就成了这几十年来的热点问题。然而，Reed-Solomon码的深洞在其译码中起重要作用，本项目旨在研究Reed-Solomon码的深洞问题，特别是投影Reed-Solomon码、扩展Reed-Solomon码和本原Reed-Solomon码，拟完全解决这三类Reed-Solomon码的深洞问题。同时，本项目还将研究MDS码的覆盖半径与深洞问题。

Coding theory is the cross discipline of mathematics and computer science. It provides methods (error correcting codes) and mathematical fundamental for reliable communications. One main task of error correcting codes is by the way of adding the least redundancy to make the message have the largest error correcting ability. The other main task is how to correct the error in the received message fast and efficiently. Since Reed-Solomon codes have advantages of good error-correcting ability and fast encoding/decoding algorithms, they become one of the earliest algebraic coding techniques applied in engineering. They have many important applications, the most prominent of which include consumer technologies such as CDs, DVDs, Blu-ray Discs, etc. The improved decoding algorithms and the limitation of their performances have been attracting the attension of many mathematicians and computer scientists for many decades. Deep holes of Reed-Solomon codes play an important role in the decoding of Reed-Solomon codes. The main issue of this project is to study the deep holes of Reed-Solomon codes, especially projective Reed-Solomon codes, extended Reed-Solomon codes and primitive Reed-Solomon codes. As an extension of our research, we will also study the covering radii and deep holes of MDS codes.