Tree-Structure Expectation Propagation for LDPC Decoding over the BEC

TL;DR

This paper introduces the Tree-EP algorithm for LDPC decoding over the BEC, which improves accuracy over belief propagation by imposing additional constraints and provides a predictive framework for finite-length performance.

Contribution

The paper develops the Tree-EP algorithm with pair-wise constraints, reformulates it as TEP for BEC, and introduces a scaling law for finite-length performance prediction.

Findings

Tree-EP yields more accurate marginal estimates than BP.

TEP decodes a higher fraction of errors than BP with similar complexity.

The scaling law accurately predicts finite-length TEP performance.

Abstract

We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). EP generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pair-wise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regime over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law (SL) for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization.