题目：On the packing density of Lee spheres

时间：2023年5月31日（周三）下午3:00-5:00

地点：北5楼四层410室

摘要：For any positive integers $n$ and $r$, the Lee sphere $S(n,r)$ of radius $r$ centered at the origin in $\Z^n$ is defined by

\[S(n,r):= \left\{(x_1,\cdots,x_n)\in \Z^n: \sum_{i=1}^n |x_i|\leq r \right\}.\]

Based on the packing density of cross-polytopes in $\R^n$, more than 50 years ago Golomb and Welch proved that the packing density of Lee spheres in $\Z^n$ tends to $0$ as $n\rightarrow \infty$ provided that the radius $r$ of the Lee sphere is large enough compared with $n$. In the same paper, they also conjectured that there is no perfect packing for $r\geq 2$ and $n\geq 3$.In this talk, we consider the lattice packing density $\delta_L(S(n,r))$ of Lee spheres. By using symmetric functions, group ring equations and etc., we can prove that $\delta_L(S(n,r))\leq \frac{2n^2+2n+1}{2n^2+2n+2}$ for $r\geq 2$ and $n\geq 3$. Moreover, several constructions of lattice packing of Lee spheres show that $\delta_L(S(n,r)) \geq \frac{2^r}{(2r+1)r!}$ as $n\rightarrow \infty$ where $r$ is fixed.

报告人简介：周悦，国防科技大学数学系 研究员。主要研究有限几何、代数组合及其在编码密码中的应用，在Adv. Math., J. Cryptology, JCTA等期刊发表论文40余篇。2016年获得国际组合及其应用学会Kirkman奖章，德国“洪堡”Fellow。2019年起担任国际期刊Designs, Codes and Cryptography编委。2021年获评国家级青年人才。