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arXiv · 2410.04800

A New Linear Programming Method in Sphere Packing

Qun Mo·Jinming Wen·Yu Xia
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Abstract

Inspired by the linear programming method developed by Cohn and Elkies (Ann. Math. 157(2): 689-714, 2003), we introduce a new linear programming method to solve the sphere packing problem. More concretely, we consider sequences of auxiliary functions $\{g_m\}_{m\in \mathbb{N}^{+}}$, where $g_m$ is a $m\Lambda$-periodic auxiliary function defined on $\mathbb{R}^n$, with $\Lambda$ being a given full-rank lattice in $\mathbb{R}^n$. This new method extends the original approach and offers a greater flexibility. Furthermore, using this new linear programming framework, we construct several effective auxiliary functions for dimensions $n=1,2,3$. We hope this approach provides valuable insights into solving sphere packing problems for $n=2,3$ and even higher dimensions.

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Qun Mo, Jinming Wen, Yu Xia. 2024-10-07. A New Linear Programming Method in Sphere Packing. https://arxiv.org/abs/2410.04800

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