Bounds for Multiple Packing and List-Decoding Error Exponents

07/12/2021 ∙ by Yihan Zhang, et al. ∙ 0

We revisit 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 ℝ^n such that any point in ℝ^n lies in the intersection of at most L-1 balls of radius √(nN) around points in 𝒞. We study the multiple packing problem for both bounded point sets whose points have norm at most √(nP) for some constant P>0 and unbounded point sets whose points are allowed to be anywhere in 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. 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) Poisson Point Processes. To this end, we apply tools from high-dimensional geometry and large deviation theory. Some of our lower bounds on the optimal multiple packing density are the best known lower bounds. These bounds are obtained via a proxy known as error exponent. The latter quantity is the best exponent of the probability of list-decoding error when the code is corrupted by a Gaussian noise. We establish a curious inequality which relates the error exponent, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We derive various bounds on the error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.



There are no comments yet.


page 42

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.