Multiple Packing: Lower Bounds via Infinite Constellations

Abstract

We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $N>0$ and $L\in\mathbb{Z}_{\ge2}$. A multiple packing is a set $\mathcal{C}$ of points in $\mathbb{R}^n$ such that any point in $\mathbb{R}^n$ lies in the intersection of at most $L-1$ balls of radius $\sqrt{nN}$ around points in $\mathcal{C}$. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive the best known lower bounds on the optimal density of list-decodable infinite constellations for constant $L$ under a stronger notion called average-radius multiple packing. To this end, we apply tools from high-dimensional geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new results added and significant revision. This paper is one of the three parts. The other two are arXiv:2211.04408 and arXiv:2211.0440