# List Decoding Random Euclidean Codes and Infinite Constellations

## TL;DR

This paper investigates the limits of list decoding for various code ensembles over real numbers, establishing bounds on list size relative to capacity and exploring properties of infinite constellations and random lattices.

## Contribution

It introduces bounds on list sizes for spherical codes and infinite constellations, and connects lattice properties with list decoding performance under adversarial conditions.

## Key findings

- Lower bounds on list size for typical random spherical codes
- Upper bounds on list size for Construction-A lattices and infinite constellations
- Polynomial growth of list size under a number-theoretic conjecture

## Abstract

We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a well-known result that when the source and the adversary have power constraints $ P $ and $ N $ respectively, the list decoding capacity is equal to $ \frac{1}{2}\log\frac{P}{N} $. Random spherical codes achieve constant list sizes, and the goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity. We show a reduction from arbitrary codes to spherical codes, and derive a lower bound on the list size of typical random spherical codes. We also give an upper bound on the list size achievable using nested Construction-A lattices and infinite Construction-A lattices. We then define and study a class of infinite constellations that generalize Construction-A lattices and prove upper and lower bounds for the same. Other goodness properties such as packing goodness and AWGN goodness of infinite constellations are proved along the way. Finally, we consider random lattices sampled from the Haar distribution and show that if a certain number-theoretic conjecture is true, then the list size grows as a polynomial function of the gap-to-capacity.