[Paper Review] Bounds for Multiple Packing and List-Decoding Error Exponents

Yihan Zhang, Shashank Vatedka|arXiv (Cornell University)|Jul 12, 2021

Mathematical Approximation and Integration57 references4 citations

TL;DR

This paper establishes new bounds on the maximum density of multiple packings in high-dimensional Euclidean space, both bounded and unbounded, using tools from high-dimensional geometry and large deviation theory. It introduces a novel connection between the error exponent in list-decoding and the worst-case list-decoding radius, yielding the best-known lower bounds on multiple packing density and exact asymptotics for average-radius list-decodable codes such as expurgated Gaussian codes.

ABSTRACT

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\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 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 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.

Motivation & Objective

• To derive tight bounds on the maximum density of multiple packings in high-dimensional Euclidean space for both bounded and unbounded point sets.
• To establish a connection between the average-case error exponent and the worst-case list-decoding radius in list-decodable codes.
• To characterize the asymptotic performance of average-radius list-decodable codes, including expurgated Gaussian codes and Poisson Point Processes.
• To improve upon existing lower bounds on multiple packing density using error exponent analysis.

Proposed method

• The authors use large deviation theory to analyze the tail behavior of list-decoding error probabilities under Gaussian noise.
• They introduce a proxy quantity—error exponent—to bound the asymptotic performance of multiple packings and list-decodable codes.
• A key technical contribution is a novel inequality linking the average-case error exponent to the worst-case list-decoding radius.
• The analysis is applied to both bounded sets (with norm ≤ √(nP)) and unbounded sets in ℝⁿ.
• The method leverages high-dimensional geometric properties, particularly concentration of measure and sphere packing analogies.
• The framework is applied to specific code ensembles, including expurgated Gaussian codes and Poisson Point Processes, to derive exact asymptotics.

Experimental results

Research questions

• RQ1 What is the optimal asymptotic density of multiple packings in high-dimensional Euclidean space under bounded and unbounded constraints?
• RQ2 How does the error exponent in list-decoding relate to the worst-case list-decoding radius in high-dimensional codes?
• RQ3 Can the error exponent framework yield tight lower bounds on multiple packing density?
• RQ4 What are the exact asymptotic performance limits of average-radius list-decodable codes such as expurgated Gaussian codes?
• RQ5 How do large deviation principles help characterize the trade-offs between code rate, list size, and error exponent?

Key findings

• The paper establishes the best-known lower bounds on the maximum density of multiple packings in high-dimensional Euclidean space.
• It derives exact asymptotics for the performance of average-radius list-decodable codes, including expurgated Gaussian codes and Poisson Point Processes.
• A novel inequality is proven that connects the average-case error exponent to the worst-case list-decoding radius, revealing a fundamental duality in coding performance.
• The error exponent analysis yields tight bounds on the asymptotic performance of list-decodable codes in both bounded and unbounded settings.
• The results provide a new analytical framework for studying list-decoding in continuous alphabets, with implications beyond multiple packing.
• The framework successfully captures the trade-off between code rate, list size, and error exponent, offering precise asymptotic limits for key code families.