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

Emmanuel Abbe, Colin Sandon, Vladyslav Shashkov, Maryna Viazovska

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.

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

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.

Paper Structure

This paper contains 19 sections, 23 theorems, 116 equations.

Table of Contents

  1. Coding problem
  2. Reed-Muller codes
  3. Main result
  4. Additive combinatorics result: orbit localization lemma
  5. Related literature
  6. Set-up for the proof of Theorem 3.1
  7. Compact notation
  8. Ruzsa distance and symmetries
  9. Freiman-Ruzsa inequality for conditional Ruzsa distance
  10. Proof of Theorem 3.1
  11. Proof of Theorem 3.1 (1)
  12. Permutation invariance
  13. Small orbit localization lemma and its corollary for subspace distance
  14. Linking permutation invariance to bounds on the entropy of sums
  15. From the recurrence bound to a polarization bound
  16. ...and 4 more sections

Key Result

Lemma 8

Consider a random variable $A$ taking values in the set $\mathcal{A}$. Define Additionally define Then, $err(A\mid B)\leq H(A\mid B)$.

Theorems & Definitions (41)

  • Definition 1: Codes that achieve capacity
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 6: RM code capacity-achieving parameter r.
  • Definition 7
  • Lemma 8
  • Theorem 9
  • Lemma 10
  • ...and 31 more