有限域上特殊多项式的置换性质和线性码的代数译码是数论和编码理论中的两个重要研究课题。本项目拟通过使用计算代数、代数数论、有限几何、组合设计等数学理论和工具，对逆 Dickson多项式的置换性质和Reed-Solomon码的译码问题展开系统、深入的研究：1、刻画置换逆Dickson多项式的充分必要条件；2、设计Reed-Solomon码的译码算法3、研究新译码算法的编码性质，包括译码算法的纠错能力、译码失败概率和数学正确性等。这些课题的研究将具有重要的理论意义和应用价值。

Permutational properties of special polynomials over finite fields and decoding linear codes are two hot topics in number theory and coding theory. Using theories and tools of computation algebra, algebraic number theory, finite geometry, and combinatorial design, this project aims to study permutation reversed Dickson polynomials and decoding problem of Reed-Solomon codes systematically and deeply. This project will study: 1. characterizing when reversed Dickson polynomials are permutations; 2. designing and improving decoding algorithms; 3. coding properties, including error correction ability, decoding failure probability and the mathematical correctness of the new algorithms. Topics studied in this project will have great applications in theory and practice.