PREPRINT

# Multiple Packing: Lower Bounds via Error Exponents

Yihan Zhang and Shashank Vatedka

Submitted on 8 November 2022, last revised on 9 November 2022

## Abstract

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 $L\in {\mathbb{Z}}_{\ge 2}$, 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 $\mathcal{C}$. 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.

## Preprint

Comment: The paper arXiv:2107.05161 has been split into three parts with new results added and significant revision. This paper is one of the three parts. The other two are arXiv:2211.04407 and arXiv:2211.04406

Subjects: Mathematics - Metric Geometry; Computer Science - Information Theory