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Capacity on BMS Channels via Code Symmetry and Nesting

Henry D. Pfister, Galen Reeves

TL;DR

The paper presents a unified, tutorial-style framework showing that binary Reed–Muller codes can achieve capacity on binary memoryless symmetric channels by leveraging nested, doubly transitive code sequences and two- (and three-) look posterior bounds. It blends decoding-function symmetry with Boolean-function Fourier analysis, hypercontractivity, and level-k inequalities to obtain exponential decay of the bit-error probability across recursion stages, yielding vanishing block-error probability near capacity on BMS channels (including the BSC). In the BEC, the approach sharpens erasure decay, while for the BSC it achieves faster decay rates via level-k and majority-vote schemes; a Transference theorem then extends these results to general BMS channels. The work thus provides a simpler, more unified proof mechanism that connects code symmetry, nesting, and functional analysis, with implications for both theory and potential future applications in structured-code design and quantum-channel contexts.

Abstract

The past decade has seen notable advances in our understanding of structured error-correcting codes, particularly binary Reed--Muller (RM) codes. While initial breakthroughs were for erasure channels based on symmetry, extending these results to the binary symmetric channel (BSC) and other binary memoryless symmetric (BMS) channels required new tools and conditions. Recent work uses nesting to obtain multiple weakly correlated "looks" that imply capacity-achieving performance under bit-MAP and block-MAP decoding. This paper revisits and extends past approaches, aiming to simplify proofs, unify insights, and remove unnecessary conditions. By leveraging powerful results from the analysis of boolean functions, we derive recursive bounds using two or three looks at each stage. This gives bounds on the bit error probability that decay exponentially in the number of stages. For the BSC, we incorporate level-k inequalities and hypercontractive techniques to achieve the faster decay rate required for vanishing block error probability. The results are presented in a semitutorial style, providing both theoretical insights and practical implications for future research on structured codes.

Capacity on BMS Channels via Code Symmetry and Nesting

TL;DR

The paper presents a unified, tutorial-style framework showing that binary Reed–Muller codes can achieve capacity on binary memoryless symmetric channels by leveraging nested, doubly transitive code sequences and two- (and three-) look posterior bounds. It blends decoding-function symmetry with Boolean-function Fourier analysis, hypercontractivity, and level-k inequalities to obtain exponential decay of the bit-error probability across recursion stages, yielding vanishing block-error probability near capacity on BMS channels (including the BSC). In the BEC, the approach sharpens erasure decay, while for the BSC it achieves faster decay rates via level-k and majority-vote schemes; a Transference theorem then extends these results to general BMS channels. The work thus provides a simpler, more unified proof mechanism that connects code symmetry, nesting, and functional analysis, with implications for both theory and potential future applications in structured-code design and quantum-channel contexts.

Abstract

The past decade has seen notable advances in our understanding of structured error-correcting codes, particularly binary Reed--Muller (RM) codes. While initial breakthroughs were for erasure channels based on symmetry, extending these results to the binary symmetric channel (BSC) and other binary memoryless symmetric (BMS) channels required new tools and conditions. Recent work uses nesting to obtain multiple weakly correlated "looks" that imply capacity-achieving performance under bit-MAP and block-MAP decoding. This paper revisits and extends past approaches, aiming to simplify proofs, unify insights, and remove unnecessary conditions. By leveraging powerful results from the analysis of boolean functions, we derive recursive bounds using two or three looks at each stage. This gives bounds on the bit error probability that decay exponentially in the number of stages. For the BSC, we incorporate level-k inequalities and hypercontractive techniques to achieve the faster decay rate required for vanishing block error probability. The results are presented in a semitutorial style, providing both theoretical insights and practical implications for future research on structured codes.

Paper Structure

This paper contains 38 sections, 29 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Outline
  4. Background
  5. Standard Notation
  6. Preliminaries
  7. Decoding Functions
  8. Erasure Channels
  9. Beyond Erasures: Multiple Looks via Nested Codes
  10. Multiple Looks, Subspace Codes, and Sunflowers
  11. Binary Memoryless Symmetric (BMS) Channels
  12. Correlation Bound for Boolean Functions
  13. Analysis for the BEC
  14. Setup
  15. Recursion
  16. ...and 23 more sections

Key Result

Theorem 1

Consider a BMS channel with capacity $C$ and a strongly nested sequence of doubly-transitive codes $\mathcal{C}_k$ with normalized overlap $\rho_k \leq \rho < 1$ for all $k\in \{0,1,\ldots\}$. If the rate of $\mathcal{C}_0$ is less than $C$, then there is a $\delta >0$ such that the bit-error probab

Figures (1)

  • Figure 1: Diagram showing the nesting structure of $\mathcal{C}$, $\mathcal{C}'$, and $\mathcal{C}"$ with sizes shown for the RM code sequence.

Theorems & Definitions (53)

  • Theorem 1: Informal BMS
  • Theorem 2: Informal Faster BSC
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 43 more