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

TL;DR

The paper analyzes the stability of LDPC codes under BP and MAP decoding across binary‑input BMS channels, establishing a stability threshold that governs universal capacity‑achieving performance. It shows that several capacity‑achieving LDPC sequences designed for the BEC are not universal for broader BMS families, with the Bhattacharyya parameter playing a central role in the analysis. By linking a density‑evolution based stability criterion to blockwise and bitwise MAP thresholds, the authors derive upper bounds on MAP thresholds and reveal the operational impact of stability on decoding performance. An exploration process and cycle‑based arguments demonstrate that cycles involving degree‑two variable nodes drive MAP decoding failures, providing concrete lower bounds on error probabilities and informing universal code design across channels. Overall, the work offers a principled criterion (stability threshold) for assessing universal LDPC constructions and highlights the tradeoffs between degree distributions and cross‑channel reliability.

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.

Figures (12)

• Figure 1: Illustration of the configuration of half-edges via $\Pi$.
• Figure 2: $\{f_{(N)}\}$ and $\{f_{(N)} ^{\prime}\}$ for the heavy-tail Poisson sequence.
• Figure 3: $\{f_{(N)} ^{\prime\prime}\}$ of the heavy-tail Poisson sequence for $\epsilon \in \{0.3, 0.5, 0.7\}$.
• Figure 4: Illustration of (\ref{['eqn:rr+F012']}) for the right-regular sequence with $\epsilon=0.5$.
• Figure 5: $\{f_{(N)} ^{\prime\prime}\}$ of the right-regular sequence for $\epsilon \in \{0.3, 0.5, 0.7\}$.

Theorems & Definitions (46)

• Example 1.1
• Example 1.2
• Example 2.1
• Lemma 2.2
• Example 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 3.1
• Theorem 3.2
• Lemma 3.3