AI 中文总结
研究在较低秩固定格平移上纤维化的球填充的线性规划界及其对偶形式，证明原问题和对偶问题能达最优解，探讨其在不同维度的情况及与已知结果的关系，还表明某些LP界不能证明特定猜想。
AI 中文摘要
我们研究了在较低秩固定格的平移上纤维化的球填充的线性规划（LP）界及其对偶形式。我们将康威和斯隆关于纤维填充的工作置于\(\mathbb{R}^k \times A\)（其中\(A\)是紧致阿贝尔群）上填充问题的线性规划界的背景下。对于这些程序，我们证明了原问题和对偶问题都能达到最优解，每个实例都有满足互补松弛性的最优对。我们研究了这里考虑的LP界在维度\(\leq 9\)且具有规定平移对称性时达到最知名球填充密度的情况，恢复了康威和斯隆的命题2、3、5和8的类似结果。我们表明巴恩斯 - 沃尔晶格对于任何在\(E_8\)平移上纤维化的16维填充都能达到最优球填充密度。在维度6中，我们表明在\(D_4\)平移上纤维化的自然LP界实际上等同于普通球填充在维度2中的LP界，而在\(A_2\)平移上纤维化的4维填充的LP界由维度2中的LP界推出，但具有额外的刚性结构，可能使其更易于处理。最后，使用李的离散约化框架，我们表明在\(A_3\)平移上纤维化的填充的线性规划界严格大于1，因此仅线性规划界不能证明科恩和拉贾戈帕尔的猜想4.1。
英文摘要
We study linear programming (LP) bounds for sphere packings that fiber over translates of a fixed lattice of lower rank, as well as their dual formulations. In doing so, we place the work of Conway and Sloane on fibered packings in the context of linear programming bounds for packing problems on $\mathbb{R}^k \times A$, where $A$ is a compact abelian group. For these programs we prove that the primal and the dual both attain their optima, so that every instance has an optimal pair satisfying complementary slackness. We study cases in which the LP bounds considered here achieve the best known sphere packing densities in dimensions $\le 9$ with prescribed translational symmetry, recovering analogues of Propositions 2, 3, 5, and 8 of Conway and Sloane, and we show that the Barnes-Wall lattice achieves the optimal sphere packing density for any $16$-dimensional packing that fibers over translates of $E_8$. In dimension $6$, we show that the natural LP bound fibering over translates of $D_4$ is in fact equivalent to the LP bound in dimension $2$ for ordinary sphere packing, while the LP bound for $4$-dimensional packings that fiber over $A_2$ translates is implied by the LP bound in dimension $2$ but has additional rigid structure that may make it more tractable. Finally, using the discrete reduction framework of Li, we show that the linear programming bound for packings fibering over translates of $A_3$ is strictly above $1$, so the linear programming bound alone cannot prove Conjecture 4.1 of Cohn and Rajagopal.
Comments27 pages
