维度 $25$--$29$ 和 $31$ 中接吻数的新下界

中文摘要 AI 辅助

维度 $d$ 中的接吻数是能同时接触一个相同大小的中心球的不重叠全等球的最大数量。我们研究维度 $25$-$31$，其中最佳先前构造基于从维度 $24$ 中的最优接吻构型进行 Leech 提升。我们的方法利用了未提升块与由提升向量和辅助向量组成的块之间缺乏接触。在保持块固定的同时旋转该块，在维度 $26$、$27$ 和 $28$ 中为两个对跖点腾出空间，在维度 $29$ 中为一个点腾出空间。两个进一步的修改在维度 $25$ 和 $31$ 中产生改进：提升块的非正交对角线性变形在维度 $25$ 中允许两个对跖点，而仅旋转提升向量的附加坐标在维度 $31$ 中允许四个非对跖点。这些构造共同产生了新的下界 $\tau_{25}\geq 197058$，$\tau_{26}\geq 198552$，$\tau_{27}\geq 200046$，$\tau_{28}\geq 204522$，$\tau_{29}\geq 209497$，以及 $\tau_{31}\geq 238354$。

英文摘要

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 $29$. Three further modifications yield improvements in dimensions $25$, $30$ and $31$: (a) a nonorthogonal diagonal linear deformation of the lifted block admits two antipodal points in dimension $25$; (b) rotating the additional coordinates of the lifted vectors and then applying a small orthogonal transformation to the resulting lifted block as a whole admits two antipodal points in dimension $30$; (c) rotating only the additional coordinates of the lifted vectors admits four nonantipodal points in dimension $31$. Together, these constructions yield the new lower bounds $τ_{25}\geq 197058$, $τ_{26}\geq 198552$, $τ_{27}\geq 200046$, $τ_{28}\geq 204522$, $τ_{29}\geq 209497$, $τ_{30}\ge 220442$, and $τ_{31}\geq 238354$.

发表机构

• Nazarbayev University（纳扎尔巴耶夫大学）
• Nazarbayev University Research Administration（纳扎尔巴耶夫大学研究行政）

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