Efficient Majority-Logic Decoding of Short-Length Reed--Muller Codes at Information Positions

TL;DR

This paper presents an improved majority-logic decoding method for short Reed--Muller codes, focusing on errors at information positions, with detailed analysis on the minimal number of majority gates and complexity comparisons.

Contribution

It introduces a more efficient decoding approach for short-length Reed--Muller codes, especially for errors at information positions, and analyzes minimal gate requirements and complexity.

Findings

Enhanced decoding efficiency for RM codes at information positions.

Detailed analysis of minimal majority gates for short codes.

Comparison of decoding complexity with existing methods.

Abstract

Short-length Reed--Muller codes under majority-logic decoding are of particular importance for efficient hardware implementations in real-time and embedded systems. This paper significantly improves Chen's two-step majority-logic decoding method for binary Reed--Muller codes $\text{RM}(r,m)$, $r \leq m/2$, if --- systematic encoding assumed --- only errors at information positions are to be corrected. Some general results on the minimal number of majority gates are presented that are particularly good for short codes. Specifically, with its importance in applications as a 3-error-correcting, self-dual code, the smallest non-trivial example, $\text{RM}(2,5)$ of dimension 16 and length 32, is investigated in detail. Further, the decoding complexity of our procedure is compared with that of Chen's decoding algorithm for various Reed--Muller codes up to length $2^{10}$.