New Sphere Packings from the Antipode Construction

TL;DR

The paper advances dense non-lattice sphere packings in seven dimensions by applying the antipode construction to carefully chosen suboptimal cross-sections of the Leech lattice $\Lambda_{24}$ and the self-dual lattice $P_{48p}$. It formalizes a special case of the antipode method, derives a center-density formula, and demonstrates that suboptimal cross-sections can outperform the corresponding densest cross-sections, yielding new records in dimensions $19$, $20$, $21$, $23$, $44$, $45$, and $47$ with explicit generator and Gram matrices. The results include concrete center densities, e.g., $\delta_{23}=0.50049\ldots$, $\delta_{21}=0.21004\ldots$, $\delta_{20}=0.15593\ldots$, $\delta_{19}=0.08896\ldots$, and for the high dimensions $\delta_{47}\approx 5925.98$, $\delta_{45}\approx 1243.46$, $\delta_{44}\approx 509.619$, illustrating substantial improvements over prior records. The work highlights the potential of the antipode construction to surpass existing cross-section translations in self-dual lattices, and it points to further dense packings in dimensions below $44$ achievable via this approach.

Abstract

In this note, we construct non-lattice sphere packings in dimensions $19$, $20$, $21$, $23$, $44$, $45$, and $47$, demonstrating record densities that surpass all previously documented results in these dimensions. The construction involves applying the antipode method to suboptimal cross-sections of $Λ_{24}$ and $P_{48p}$ respectively in those dimensions.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• proof