Maxwell Construction: The Hidden Bridge between Iterative and Maximum a Posteriori Decoding

TL;DR

This paper introduces the Maxwell decoder, revealing a fundamental link between belief propagation and MAP decoding, with detailed analysis for binary erasure channels and potential extensions to other channels.

Contribution

The paper presents the Maxwell decoder, a novel algorithm that explicitly connects belief propagation and MAP decoding, inspired by thermodynamic Maxwell construction.

Findings

The Maxwell decoder provides a constructive link between belief propagation and MAP decoding.

Detailed analysis of the Maxwell decoder for binary erasure channels.

Discussion of extensions to general binary memoryless channels.

Abstract

There is a fundamental relationship between belief propagation and maximum a posteriori decoding. A decoding algorithm, which we call the Maxwell decoder, is introduced and provides a constructive description of this relationship. Both, the algorithm itself and the analysis of the new decoder are reminiscent of the Maxwell construction in thermodynamics. This paper investigates in detail the case of transmission over the binary erasure channel, while the extension to general binary memoryless channels is discussed in a companion paper.