Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912924459597824 |
|---|---|
| author | Abbe, Emmanuel Sandon, Colin Shashkov, Vladyslav Viazovska, Maryna |
| author_facet | Abbe, Emmanuel Sandon, Colin Shashkov, Vladyslav Viazovska, Maryna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13493 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity Abbe, Emmanuel Sandon, Colin Shashkov, Vladyslav Viazovska, Maryna Information Theory Combinatorics Number Theory 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. |
| title | Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity |
| topic | Information Theory Combinatorics Number Theory |
| url | https://arxiv.org/abs/2411.13493 |