Reed-Muller码是一类非常重要的纠错码，被用于深空通信中。Reed-Muller码在理论计算机科学中有着重要应用，比如概率检验证明、低度数测试、局部测试等。Reed-Muller码本身就是一类密码函数的集合，研究其译码算法及深洞问题将对密码函数的研究有着重要的应用前景，如：一阶Reed-Muller码的深洞就对应于Bent函数，高阶Reed-Muller码的深洞就对应于具有高阶代数免疫度的密码函数，这些算法都会对寻找具有特定代数免疫的密码函数起到重要的启发作用等等。所以研究其内在结构以及译码算法具有非常重要的理论和应用价值，该项目将主要研究Reed-Muller码的重量分布、列表译码及相关问题，拟推进或解决Reed-Muller码的列表译码问题、覆盖半径问题、深洞问题等。

Reed-Muller codes are very important error-correcting codes, which are used in deep space communications such as early Mars exploration. Reed-Muller codes have important applications in theoretical computer science, such as probability test proof, low-degree test, local test and so on. Reed-Muller code itself is a set of cryptographic functions. Studying its decoding algorithm and deep hole problem will have important application prospects for the research of cryptographic functions. For example, the deep holes of first-order Reed-Muller code correspond to Bent functions, and the deep holes of higher-order Reed-Muller code correspond to the cryptographic functions with higher-order algebraic immunity. These algorithms will play an important role in finding cryptographic functions with certain algebraic immunity. Therefore, the study of its internal structure and decoding algorithm has very important theoretical and practical value. This project will mainly study the weight distribution, list decoding and related problems of Reed-Muller codes, and will improve or completely solve the list decoding, covering radius and deep hole problems of Reed-Muller codes.