List-decodability of Poisson Point Processes

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∈Z≥2. A multiple packing is a set C of points in Rn such that any point in Rn lies in the intersection of at most L – 1 balls of radius nN−−−√ around points in 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 exactly pin down the asymptotic density of (expurgated) Poisson Point Processes under a stronger notion called average-radius multiple packing. To this end, we apply tools from high-dimensional geometry and large deviation theory. This gives rise to the best known lower bound on the largest multiple packing density. Our result corrects a mistake in a previous paper by Blinovsky [Bli05].