From Bit to Block: Decoding on Erasure Channels

TL;DR

This paper introduces a general framework to relate bit and block error thresholds for linear codes over erasure channels, using subcode support weights, and demonstrates its effectiveness with Reed-Muller codes achieving capacity.

Contribution

It provides a novel theoretical approach linking bit and block error thresholds via subcode support weights, and offers a new proof of Reed-Muller codes achieving capacity.

Findings

Framework bounds block error threshold using subcode support weights

New proof that Reed-Muller codes achieve capacity on erasure channels

Method applicable to analyzing other linear codes

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.