最优N-重线性码的构造及其t-设计研究

线性码由于具有良好的代数结构以及易于描述和加解密的特性, 在通信、数据存储和信息安全等领域具有重要的应用, 并且一直是数学和经典的纠错码理论研究的重点. 本项目拟综合利用有限域、组合设计、数论等理论和工具, 对具有 N-重的线性码的设计以及可约循环码的若干问题展开深入研究, 具体包括: (1) 尝试利用零差分平衡函数和准完全非线性函数等新的密码函数来设计线性码, 并讨论码的最小距离、维数、重量分布以及完全重量枚举等重要指标. (2) 研究基于新型分圆理论的线性码和循环码, 尤其是具有最优参数的线性码和循环码. (3) 利用线性码和t-设计的关系, 探讨利用具有很少非零重量的线性码构造t-设计. 这些课题的研究具有重要的理论意义和应用价值.

Due to their sound algebraic structure and efficient encoding and decoding algorithms，linear codes have important applications in communication, data storage and information security, as well as the focus of mathematical and classical error correction theory. The main objectives of this project are to give a well-rounded treatment of linear code with N-weight and reducible cyclic code based on the theory of finite fields, exponential sums, combinatorial design, and number theory, which include: (1) Try to use zero differential balance function and near-perfect nonlinear functions to construct linear codes and discuss the minimum distance, dimension, weight distribution and complete weight enumerators of the code. (2) Research on the linear codes and cyclic codes based on new cyclotomic theory, especially the codes with optimal parameters. (3) With the relationship between linear codes and t-design, we will construct the t-design using the linear code with a few nonzero weights. Research on these subjects has both important theoretical significance and application value.