Multiple Packing:Lower Bounds via Infinite Constellations

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.