  We derive lower bounds on the maximal rates for multiple packings in
high-dimensional Euclidean spaces. Multiple packing is a natural generalization
of the sphere packing problem. For any $ 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} $. We study this problem
for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some
constant $P>0$ and unbounded point sets whose points are allowed to be anywhere
in $ \mathbb{R}^n $. 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. We derive the best known lower bounds on
the optimal multiple packing density. This is accomplished by establishing a
curious inequality which relates the list-decoding error exponent for additive
white Gaussian noise channels, a quantity of average-case nature, to the
list-decoding radius, a quantity of worst-case nature. We also derive various
bounds on the list-decoding error exponent in both bounded and unbounded
settings which are of independent interest beyond multiple packing.
