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

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 ParityCheck Codes Under Message-Passing Decoding’ by Thomas J. Richardson and Rüdiger L. Urbanke[1]. We further provide a new proof of that theorem based on a different argument.