# Recursive Decoding and Its Performance for Low-Rate Reed-Muller Codes

## TL;DR

This paper introduces a recursive decoding algorithm for Reed-Muller codes that achieves near-optimal error correction with subexponential complexity, improving known asymptotic bounds for decoding performance.

## Contribution

The paper presents a new recursive decoding algorithm for Reed-Muller codes with complexity $n ext{log} n$ that corrects most errors up to a certain weight, surpassing previous bounds.

## Key findings

- Algorithm corrects most error patterns of weight up to $n(1/2- ext{epsilon})$
- Decoding complexity is of order $n ext{log} n$
- Improves asymptotic bounds for decoding Reed-Muller codes

## Abstract

Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length $n$ and fixed order $r.$ An algorithm is designed that has complexity of order $n\log n$ and corrects most error patterns of weight up to $n(1/2-\varepsilon)$ given that $\varepsilon$ exceeds $n^{-1/2^{r}}.$ This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity.