# Soft decision decoding of Reed-Muller codes: recursive lists

## TL;DR

This paper introduces an improved recursive list decoding algorithm for Reed-Muller codes that approaches maximum-likelihood performance, utilizing permutation techniques and error-prone bit elimination, with extensive simulation results.

## Contribution

The paper presents a novel recursive list decoding method for Reed-Muller codes that significantly improves decoding performance and provides tight bounds on ML performance.

## Key findings

- Approaches ML performance within 0.1 dB for length 256 codes.
- Effective use of permutation techniques enhances decoding accuracy.
- Elimination of error-prone bits improves overall decoding performance.

## Abstract

Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes.