A Local Tree Structure is NOT Sufficient for the Local Optimality of Message-Passing Decoding in Low Density Parity Check Codes

TL;DR

This paper demonstrates that a local tree structure alone does not guarantee local optimality in message-passing decoding of LDPC codes, identifies a flaw in a previous proof, and offers a new proof for the theorem.

Contribution

It shows the insufficiency of local tree structures for optimality, corrects a flaw in prior work, and provides an alternative proof of a key theorem.

Findings

Local tree structures are not sufficient for local optimality in message-passing decoding.

Identified a flaw in the proof of a fundamental theorem by Richardson and Urbanke.

Provided a new proof of the theorem based on a different approach.

Abstract

We address the problem,`Is a local tree structure sufficient for the local optimality of message passing algorithm in low density parity check codes?'.It is shown that the answer is negative. Using this observation, we pinpoint a flaw in the proof of Theorem 1 in the paper `The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding' by Thomas J. Richardson and R\"udiger L.Urbanke\cite{RUCapacity}. We further provide a new proof of that theorem based on a different argument.