A New Linear Programming Method in Sphere Packing

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 $m\hat{I}›$-periodic auxiliary function defined on ${\mathbb{R}}^n$, with $\hat{I}›$ 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.