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 LβZβ₯2β. A multiple packing is a
set C of points in Rn such that any point in Rn lies in the intersection of at most Lβ1 balls of radius nNβ around points in C. We study the multiple packing
problem for both bounded point sets whose points have norm at most nPβ
for some constant P>0 and unbounded point sets whose points are allowed to be
anywhere in Rn. 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
various bounds on the largest possible density of a multiple packing in both
bounded and unbounded settings. A related notion called average-radius multiple
packing is also studied. Some of our lower bounds exactly pin down the
asymptotics of certain ensembles of average-radius list-decodable codes, e.g.,
(expurgated) Gaussian codes and (expurgated) spherical codes. In particular,
our lower bound obtained from spherical codes is the best known lower bound on
the optimal multiple packing density and is the first lower bound that
approaches the known large L limit under the average-radius notion of
multiple packing. To derive these results, 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