Multiple Packing: Lower Bounds via Error Exponents

We derive lower bounds on the maximal rates for multiple packings in high-dimensional Euclidean spaces. For any
$ 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} $
. This is a natural generalization of the sphere packing problem. We study the multiple packing 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 over finite fields. We derive the best known lower bounds on the optimal multiple packing density. This is accomplished by establishing an 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 novel bounds on the list-decoding error exponent for infinite constellations and closed-form expressions for the list-decoding error exponents for the power-constrained AWGN channel, which may be of independent interest beyond multiple packing.