基于对映体构造的39维和43维球体填充
Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43
- State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（处理器重点实验室，中国科学院计算技术研究所）
- University of Chinese Academy of Sciences（中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究在39维和43维构造非格球体填充，通过对映体构造应用于48维晶格横截面实现，改进了记录，43维填充提高了接吻数，39维填充有新的十点对映体簇，且提供了机器可检查证书等。
AI中文摘要：
我们在39维和43维构造了非格球体填充，改进了自1982年康威和斯隆的层压晶格构造以来一直保持的记录。这两种填充均来自应用于极值偶幺模48维晶格横截面的对映体构造。43维填充还提高了已知的最佳接吻数。39维填充基于康威和斯隆在1982年使用的横截面；只是在其上放置的十点对映体簇是新的。所有声明的机器可检查证书以及搜索和验证代码均公开可用。
英文摘要：
We improve the best known sphere packings in every dimension from $38$ to $43$. Our packings in dimensions $38$ to $42$ come from one chain of cross-sections of the extremal even unimodular lattice $P_{48p}$, $\sqrt3E_6 \subset \sqrt3E_7 \subset \sqrt3E_8 \subset K_9 \subset K_{10}$, whose first three members are cut from the fixed lattice $\sqrt3(E_8 \perp E_8)$ of an order-three automorphism; their orthogonal complements are lattice packings and set the records in dimensions $42$ down to $38$. Each successive section is a determinant-minimal extension of its predecessor. Our $43$-dimensional packing is an antipode packing: six translates of the complement of a $5$-dimensional section. We also improve some kissing numbers. Conway and Sloane's twelve $1982$ cross-section packings appear never to have had theirs computed; we compute them and find that, in dimensions $42$ to $47$, they exceed the previously tabulated lower bounds. Our chain does better in dimensions $40$, $41$ and $42$, and a further antipode packing beats the record in dimension $45$.
