# The Necessity of Scheduling in Compute-and-Forward

## TL;DR

This paper analyzes the limitations of compute-and-forward (CF) in large networks, showing that without scheduling, CF degenerates to decoding only the strongest user, leading to zero sum-rate, and demonstrates scheduling's importance to maintain performance.

## Contribution

The paper reveals the asymptotic behavior of CF as the number of transmitters grows and highlights the necessity of scheduling to preserve CF's advantages in large systems.

## Key findings

- CF degenerates to decoding only the strongest user with many transmitters
- Sum-rate tends to zero without scheduling in large networks
- Scheduling enables non-trivial linear combinations and maintains sum-rate

## Abstract

Compute and Forward (CF) is a promising relaying scheme which, instead of decoding single messages or forwarding/amplifying information at the relay, decodes linear combinations of the simultaneously transmitted messages. The current literature includes several coding schemes and results on the degrees of freedom in CF, yet for systems with a fixed number of transmitters and receivers. It is unclear, however, how CF behaves at the limit of a large number of transmitters. In this paper, we investigate the performance of CF in that regime. Specifically, we show that as the number of transmitters grows, CF becomes degenerated, in the sense that a relay prefers to decode only one (strongest) user instead of any other linear combination of the transmitted codewords, treating the other users as noise. Moreover, the sum-rate tends to zero as well. This makes scheduling necessary in order to maintain the superior abilities CF provides. Indeed, under scheduling, we show that non-trivial linear combinations are chosen, and the sum-rate does not decay, even without state information at the transmitters and without interference alignment.