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.