两种形状的格镶嵌问题研究

Problems of lattice tilings have a long history, which were first introduced by Minkowski. This project is devoted to investigating lattice tilings by two different shapes: quasi-crosses and Lee spheres. Lattice tilings by quasi-crosses are equivalent to perfect splitter sets. Non-singular perfect splitter sets have been well studied, while there are only a few results on singular perfect splitter sets. For the lattice tilings by Lee spheres, Golomb and Welch conjectured that there does not exist such tilings except for trivial cases. Moreover, since there are only a few perfect splitter sets and perfect Lee codes, constructions of quasi perfect splitter sets and quasi perfect Lee codes make sense. In this project, by combining together the theory and methods in algebra, number theory and finite geometry, we plan to make great improvements in the following specific problems: nonexistence of singular perfect splitter sets; constructions of quasi perfect splitter sets; Golomb-Welch conjecture and constructions of quasi perfect Lee codes.

格镶嵌问题是一个有着悠久历史的数学问题，它最早是由Minkowski提出来的。本项目拟研究两种形状的格镶嵌问题：准十字形和Lee球。准十字形的格镶嵌等价于完美分解集。目前，大部分非奇异完美分解集都有比较好的刻画，但奇异完美分解集的研究比较少。而对于Lee球的格镶嵌问题，Golomb和Welch猜测：除了平凡情形，Lee球的格镶嵌均不存在。由于完美分解集和完美Lee码非常稀少，因此有必要构造准完美分解集和准完美Lee码。本项目拟结合代数、数论和有限几何的理论方法，在以下具体问题上取得重要进展：奇异完美分解集的不存在性；准完美分解集的构造；Golomb-Welch猜想和准完美Lee码的构造。

在项目资助期间，申请人聚焦于利用代数和数论的方法研究组合构型的存在性问题，主要包括极值组合、离散几何和代数编码。极值组合方面，延拓和发展了随机代数构造方法。离散几何方面，引入了新的群环工具，开创了统一证明半径为2时Golomb-Welch猜想成立的新方法。代数编码方面，给出了一批性能更优的子空间码。在SCM、JCTA、SIDMA、IEEE-TIT等国际重要期刊上发表论文8篇。