格和编码理论中若干问题研究

格与球填充问题有着悠久的历史，自1611年开普勒提出著名的开普勒猜想至今已有400余年。格与球填充问题不仅是离散几何的核心研究内容，也是计算复杂性理论的重点研究课题。编码理论是研究信息传输过程中信号编码规律的数学理论。编码理论与信息论、组合数学、数论、有限域等学科都有着非常密切的联系，已成为应用数学的一个分支。本项目将综合运用模形式理论、特征和、代数组合学、线性规划、半正定规划等数学理论工具，以及MAGMA、lrslib、Julia等多种计算软件来推进强完美格的完全分类、Reed-Muller码的重量分布、LRC码的界与构造等问题。本项目的研究内容有着广泛的应用背景，研究方法涉及到数学领域的多个分支，因此具有重要的理论价值和实际意义。

The problem of lattices and sphere packings dates back to 1611 when Kepler proposed the famous Kepler conjecture. This problem is not only the core subject in discrete geometry, but also a key topic in theoretical complexity theory. Coding theory is the mathematical theory of encoding and decoding signals for the transmission of information. Coding theory has close connections to information theory, combinatorics, number theory and finite fields, and it has become a branch of applied mathematics. In this project, we will use the mathematical tools from modular forms, character theory, algebraic combinatorics, linear programming, semi-definite programming, and computational softwares such as MAGMA, lrslib, Julia, to explore the classification of strongly perfect lattices, the weight distribution of Reed-Muller codes and the bounds and the construction for LRC codes. The research content of this project is widely used and the methods involved touch on many branches in mathematics, so it has important theoretical value and practical significance.