Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity

TL;DR

This paper establishes a polarization theory for Reed-Muller codes, connecting it with recent advances in additive combinatorics and entropy extraction, demonstrating RM codes' capacity-achieving properties.

Contribution

It introduces a polarization framework for RM codes, linking it to the Polynomial Freiman-Ruzsa conjecture and entropy methods, advancing understanding of their capacity performance.

Findings

RM codes have vanishing local error below capacity

Established a connection with Polynomial Freiman-Ruzsa conjecture

Proposed a new additive combinatorics conjecture

Abstract

In 1948, Shannon used a probabilistic argument to show the existence of codes achieving a maximal rate defined by the channel capacity. In 1954, Muller and Reed introduced a simple deterministic code construction based on polynomial evaluations, which was conjectured and eventually proven to achieve capacity. Meanwhile, polarization theory emerged as an analytic framework to prove capacity results for a variation of RM codes - the polar codes. Polarization theory further gave a powerful framework for various other code constructions, but it remained unfulfilled for RM codes. In this paper, we settle the establishment of a polarization theory for RM codes, which implies in particular that RM codes have a vanishing local error below capacity. Our proof puts forward a striking connection with the recent proof of the Polynomial Freiman-Ruzsa conjecture [40] and an entropy extraction approach related to [2]. It further puts forward a small orbit localization lemma of potential broader applicability in combinatorial number theory. Finally, a new additive combinatorics conjecture is put forward, with potentially broader applications to coding theory.