# Capacity per Unit-Energy of Gaussian Many-Access Channels

## TL;DR

This paper analyzes the capacity per unit-energy in Gaussian many-access channels with a growing number of users, revealing thresholds for achievable rates and the suboptimality of orthogonal codebooks.

## Contribution

It establishes the critical user growth rate relative to blocklength for positive capacity per unit-energy and shows orthogonal schemes can be suboptimal.

## Key findings

- Users cannot achieve positive rate per unit-energy if user count exceeds n/log n.
- Single-user capacity per unit-energy is achievable below this user growth rate using orthogonal schemes.
- Orthogonal codebooks may be suboptimal when the number of users grows with blocklength.

## Abstract

We consider a Gaussian multiple-access channel where the number of transmitters grows with the blocklength $n$. For this setup, the maximum number of bits that can be transmitted reliably per unit-energy is analyzed. We show that if the number of users is of an order strictly above $n/\log n$, then the users cannot achieve any positive rate per unit-energy. In contrast, if the number of users is of order strictly below $n/\log n$, then each user can achieve the single-user capacity per unit-energy $(\log e)/N_0$ (where $N_0/ 2$ is the noise power) by using an orthogonal access scheme such as time division multiple access. We further demonstrate that orthogonal codebooks, which achieve the capacity per unit-energy when the number of users is bounded, can be strictly suboptimal.