# On the Necessity of Structured Codes for Communications over MAC with Feedback

## TL;DR

This paper introduces a new coding strategy for three-user MAC with feedback, utilizing quasi-linear codes to achieve a larger rate region than previous schemes, and demonstrates the necessity of structured codes for optimality.

## Contribution

It presents a novel coding scheme based on quasi-linear codes that extends the Cover-Leung scheme and proves the importance of algebraic structure for optimal communication.

## Key findings

- The new scheme achieves a strictly larger rate region than the Cover-Leung scheme.
- Optimal schemes for the example require codebooks to be closed under binary addition.
- The proposed coding strategy achieves optimality in certain cases.

## Abstract

The problem of three-user multiple-access channel (MAC) with noiseless feedback is investigated. A new coding strategy is presented. The coding scheme builds upon the natural extension of the Cover-Leung (CL) scheme; and uses quasi-linear codes. A new single-letter achievable rate region is derived. The new achievable region strictly contains the CL region. This is shown through an example. In this example, the coding scheme achieves optimality in terms of transmission rates. It is shown that any optimality achieving scheme for this example must have a specific algebraic structure. Particularly, the codebooks must be closed under binary addition.