🤖 AI Summary
本文研究了25-31维空间中的接吻数问题，通过旋转和变形等方法改进了基于Leech提升的最优配置，提高了这些维度下的下界。
📝 Abstract
The kissing number in dimension $d$ is the largest number of non-overlapping congruent spheres that can simultaneously touch a central sphere of the same size. We study dimensions $25$-$31$, where the best previous constructions are based on Leech lifting from the optimal kissing configuration in dimension $24$. Our method exploits the absence of contacts between the unlifted bulk and the block consisting of lifted and auxiliary vectors. Rotating this block while keeping the bulk fixed creates room for two antipodal points in dimensions $26$, $27$, and $28$, and one point in dimension $29$.
Two further modifications yield improvements in dimensions $25$ and $31$: a nonorthogonal diagonal linear deformation of the lifted block admits two antipodal points in dimension $25$, while rotating only the additional coordinates of the lifted vectors admits four nonantipodal points in dimension $31$. Together, these constructions yield the new lower bounds $τ_{25}\geq 197058$, $τ_{26}\geq 198552$, $τ_{27}\geq 200046$, $τ_{28}\geq 204522$, $τ_{29}\geq 209497$, and $τ_{31}\geq 238354$.