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Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes

Hoang Ly, Emina Soljanin

TL;DR

The paper introduces the first one-step majority-logic decoder for binary Reed–Muller codes that applies uniformly to all $(r,m)$. By exploiting a geometric recovery-set structure tied to $ ext{EG}(m,2)$, it proves the decoder can correct up to $d_{ extmin}/4$ errors and up to $d_{ extmin}-1$ erasures, and it shows these limits are tight within the 1S-MLD paradigm. The approach generalizes Reed’s original algorithm into a parallel, single-step process and reveals deep connections between recovery-set geometry and coding-theoretic limits, including a new transversal theorem for truncated flats. These results offer a practical, low-latency decoding option for RM codes and point to extensions to more general code families.

Abstract

The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability.

Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes

TL;DR

The paper introduces the first one-step majority-logic decoder for binary Reed–Muller codes that applies uniformly to all . By exploiting a geometric recovery-set structure tied to , it proves the decoder can correct up to errors and up to erasures, and it shows these limits are tight within the 1S-MLD paradigm. The approach generalizes Reed’s original algorithm into a parallel, single-step process and reveals deep connections between recovery-set geometry and coding-theoretic limits, including a new transversal theorem for truncated flats. These results offer a practical, low-latency decoding option for RM codes and point to extensions to more general code families.

Abstract

The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability.

Paper Structure

This paper contains 10 sections, 11 theorems, 27 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Basic Definitions and Notations
  3. Reed--Muller Codes Preliminaries
  4. Reed--Muller Codes
  5. A Geometric View
  6. Reed's Decoding Algorithm
  7. 1-step Majority-logic Decoding
  8. Recovery Sets of All Message Symbols
  9. One-step Majority-logic Decoding of RM codes
  10. Conclusion

Key Result

Theorem 1

Let $f$ be a minimum weight codeword of $\mathrm{RM}(r, m)$, say $f = \boldsymbol{\chi}(S)$. Then $S$ is an $(m-r)$-dimensional flat in $\mathrm{EG}(m, 2)$. Equivalently, the minimum-weight codewords of $\mathrm{RM}(r, m)$ are precisely the incidence vectors of the $(m - r)$-flats in $\mathrm{EG}(m,

Figures (1)

  • Figure 1: A schematic “flower” structure: the kernel (red) is the 1-dimensional subspace $\mathscr{S}$, while each petal (blue) represents a recovery set $S_j$ formed as the complement of $\mathscr{S}$ in a 3-dimensional subspace.

Theorems & Definitions (21)

  • Definition 1: see, e.g.,Coding:books/MacWilliamsS77
  • Theorem 1: see, e.g., Coding:books/MacWilliamsS77
  • Theorem 2: see, e.g., Coding:books/PetersonW72
  • Proposition 1: Universal limitation for 1S-MLD on RM codes
  • proof
  • Remark 1
  • Theorem 3: Reed's decoding algorithm
  • Theorem 4: SRR_RM_ISITSRR:preprint/arxiv/LySL25
  • Theorem 5: SRR_RM_ISITSRR:preprint/arxiv/LySL25
  • Theorem 6: SRR_RM_ISITSRR:preprint/arxiv/LySL25
  • ...and 11 more