低维稠格
Dense lattices in low dimensions
摘要：
巴尼斯-沃尔晶格${\bf Λ}_{16}$的中心密度为${\{1}{16}}$，吻数为4320，发现于1959年，是唯一已知的在16维中的密度球堆积。J. H.康威和新泽西Sloane证明了${\bf Λ}_{16}$是16维的密排晶格。有时候，Barnes-Wall格${\bf Λ}_{16}$是唯一的密排格，是${\bf R}^{16}$中的最佳球填充。本文构造了两个中心密度为$\{1}{16}$，接吻数为4224和4176的16维格.这导致了几个新的14和15维晶格，它们具有相同的中心密度，但不同的接吻数作为目前已知的在这两个维度中的dennial晶格。这给出了一个否定的答案，长期以来的期望，${\bf Λ}_n$的，$n \leq 24, n \neq 11,12,13$是唯一的denatrium晶格在其尺寸。【摘要】
英文摘要：
The Barnes-Wall lattice ${\bf Λ}_{16}$ with the center density ${\{1}{16}}$ and the kissing number 4320 was found in 1959 and is the only known densest sphere packing in the dimension 16. J. H. Conway and N.J.A. Sloane conjectured that ${\bf Λ}_{16}$ is the densest 16 dimensional lattice. Sometimes it is conjectured that the Barnes-Wall lattice ${\bf Λ}_{16}$ is the only densest lattice and the optimal sphere packing in ${\bf R}^{16}$. In this paper two new 16 dimensional lattices with the center density $\{1}{16}$ and the kissing numbers 4224 and 4176 are constructed. This leads to several new 14 and 15 dimensional lattices which have the same center densities but different kissing numbers as the presently known densest lattices in these two dimensions. This gives a negative answer to the long time expectation that ${\bf Λ}_n$'s, $n \leq 24, n \neq 11,12,13$ are the only densest lattices in their dimensions. {abstract}
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