The Stability of Low-Density Parity-Check Codes and Some of Its Consequences

TL;DR

This paper investigates the stability of LDPC codes under MAP decoding over symmetric channels, revealing that many capacity-achieving sequences are not universal under belief propagation decoding due to stability constraints.

Contribution

It introduces a stability threshold for LDPC codes that predicts non-universality under BP decoding and applies to MAP decoding, providing new insights into code performance limits.

Findings

Many capacity-achieving LDPC sequences are not universal under BP decoding.

The stability threshold determines an upper bound on MAP decoding thresholds.

Stability analysis links code properties to decoding performance limits.

Abstract

We study the stability of low-density parity-check (LDPC) codes under blockwise or bitwise maximum $\textit{a posteriori}$ (MAP) decoding, where transmission takes place over a binary-input memoryless output-symmetric channel. Our study stems from the consideration of constructing universal capacity-achieving codes under low-complexity decoding algorithms, where universality refers to the fact that we are considering a family of channels with equal capacity. Consider, e.g., the right-regular sequence by Shokrollahi and the heavy-tail Poisson sequence by Luby $\textit{et al}$. Both sequences are provably capacity-achieving under belief propagation (BP) decoding when transmission takes place over the binary erasure channel (BEC). In this paper we show that many existing capacity-achieving sequences of LDPC codes are not universal under BP decoding. We reveal that the key to showing this non-universality result is determined by the stability of the underlying codes. More concretely, for an ordered and complete channel family and a sequence of LDPC code ensembles, we determine a stability threshold associated with them, which further gives rise to a sufficient condition such that the sequence is not universal under BP decoding. Furthermore, we show that the same stability threshold applies to blockwise or bitwise MAP decoding as well. We present how stability can determine an upper bound on the corresponding blockwise or bitwise MAP threshold, revealing the operational significance of the stability threshold.