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

Emmanuel Abbe; Colin Sandon; Vladyslav Shashkov; Maryna Viazovska

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arXiv · 2411.13493

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

Emmanuel Abbe·Colin Sandon·Vladyslav Shashkov·Maryna Viazovska

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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.

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Emmanuel Abbe, Colin Sandon, Vladyslav Shashkov, Maryna Viazovska. 2024-11-20. Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity. https://arxiv.org/abs/2411.13493

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