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From Bit to Block: Decoding on Erasure Channels

Henry D. Pfister, Oscar Sprumont, Gilles Zémor

TL;DR

This work develops a general framework to bound the gap between a linear code’s bit- and block-error thresholds on the erasure channel by analyzing the minimum support weight $d_r(C)$ of its $r$-dimensional subcodes. Leveraging sharp-threshold techniques (Tillich–Zémor) and Margulis–Russo-type lemmas, the authors bound how close the block-MAP threshold can be to the bit-MAP threshold under conditions on subcode support growth, and show that doubly transitive codes with sufficiently large $d_r(C)$ have near-capacity block performance. As a proof of concept, Reed–Muller codes of constant rate are shown to satisfy these subcode-growth conditions, with Wei’s subcodes providing explicit lower bounds on $d_r(C)$; this yields an alternative, $d_r$-based proof that RM codes achieve capacity on the erasure channel under block-MAP decoding. The results offer a modular pathway to extend capacity results to other code families by bounding minimum-support weights of small-dimension subcodes. Overall, the paper connects structural code properties to threshold behavior in erasure channels, bridging bit- and block-error analyses with potential broad applicability.

Abstract

We provide a general framework for bounding the block error threshold of a linear code $C\subseteq \mathbb{F}_2^N$ over the erasure channel in terms of its bit error threshold. Our approach relies on understanding the minimum support weight of any $r$-dimensional subcode of $C$, for all small values of $r$. As a proof of concept, we use our machinery to obtain a new proof of the celebrated result that Reed-Muller codes achieve capacity on the erasure channel with respect to block error probability.

From Bit to Block: Decoding on Erasure Channels

TL;DR

This work develops a general framework to bound the gap between a linear code’s bit- and block-error thresholds on the erasure channel by analyzing the minimum support weight of its -dimensional subcodes. Leveraging sharp-threshold techniques (Tillich–Zémor) and Margulis–Russo-type lemmas, the authors bound how close the block-MAP threshold can be to the bit-MAP threshold under conditions on subcode support growth, and show that doubly transitive codes with sufficiently large have near-capacity block performance. As a proof of concept, Reed–Muller codes of constant rate are shown to satisfy these subcode-growth conditions, with Wei’s subcodes providing explicit lower bounds on ; this yields an alternative, -based proof that RM codes achieve capacity on the erasure channel under block-MAP decoding. The results offer a modular pathway to extend capacity results to other code families by bounding minimum-support weights of small-dimension subcodes. Overall, the paper connects structural code properties to threshold behavior in erasure channels, bridging bit- and block-error analyses with potential broad applicability.

Abstract

We provide a general framework for bounding the block error threshold of a linear code over the erasure channel in terms of its bit error threshold. Our approach relies on understanding the minimum support weight of any -dimensional subcode of , for all small values of . As a proof of concept, we use our machinery to obtain a new proof of the celebrated result that Reed-Muller codes achieve capacity on the erasure channel with respect to block error probability.
Paper Structure (9 sections, 11 theorems, 65 equations)

This paper contains 9 sections, 11 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Techniques
  4. Preliminaries
  5. Notation and Conventions
  6. Known Results
  7. General Linear Codes
  8. Doubly Transitive Codes
  9. Reed--Muller Codes

Key Result

Theorem 1

Consider any linear code $C\subseteq\mathbb{F}_2^N$ and suppose that a uniform random codeword $c\in C$ is sent over the erasure channel with erasure probability $p\in[0,1]$. Let $\delta\in[0,1]$ be such that, for every $i\in [N]$, the probability we fail to decode the $i^\textnormal{th}$ coordinate Suppose additionally that for all $r=1,2,\dotsc, \sqrt{\delta}N$, the support of any $r$-dimensiona

Theorems & Definitions (19)

  • Theorem 1: Informal
  • Theorem 2: follows from kudekar2016erasure, Lemma 34 and Theorem 19
  • Theorem 3: follows from wei1991RMsubcodes, Theorem 7
  • Lemma 4: Margulis-Russo Lemma, margulis1974transitionrusso1982transition
  • Lemma 5: follows from tillich2000sharptransition, page 476
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Theorem 8
  • ...and 9 more