Multiple Packing:Lower Bounds via Infinite Constellations

Yihan Zhang, Shashank Vatedka

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}_{\ge 2} $ . 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.