Density Evolution Analysis of Generalized Low-density Parity-check Codes under a Posteriori Probability Decoder

TL;DR

The paper analyzes generalized LDPC codes under an APP decoder, extending density-evolution analysis to GLDPC ensembles on binary memoryless symmetric channels. It introduces GLDPC design with a GC-node fraction $t$, proves concentration, symmetry, and monotonicity properties to justify density evolution, and develops both BEC and BI-AWGN analytic frameworks. A key contribution is the notion of message-invariant subcodes, which simplifies both analysis and decoding, complemented by a Gaussian-mixture approach that substantially improves BI-AWGN thresholds while preserving low complexity. Through simulations, GLDPC codes with suitably chosen GC-proportions achieve significant performance gains over base LDPC codes at the same design rate. The work provides practical design tools to push GLDPC codes closer to capacity with manageable decoding complexity, and suggests directions for subcode selection and error-floor investigation.

Abstract

In this study, the performance of generalized low-density parity-check (GLDPC) codes under the a posteriori probability (APP) decoder is analyzed. We explore the concentration, symmetry, and monotonicity properties of GLDPC codes under the APP decoder, extending the applicability of density evolution to GLDPC codes. On the binary memoryless symmetric channels, using the BEC and BI-AWGN channels as two examples, we demonstrate that with an appropriate proportion of generalized constraint (GC) nodes, GLDPC codes can reduce the original gap to capacity compared to their original LDPC counterparts. Additionally, on the BI-AWGN channel, we apply and improve the Gaussian approximation algorithm in the density evolution of GLDPC codes. By adopting Gaussian mixture distributions to approximate the message distributions from variable nodes and Gaussian distributions for those from constraint nodes, the precision of the channel parameter threshold can be significantly enhanced while maintaining a low computational complexity similar to that of Gaussian approximations. Furthermore, we identify a class of subcodes that can greatly simplify the performance analysis and practical decoding of GLDPC codes, which we refer to as message-invariant subcodes. Using the aforementioned techniques, our simulation experiments provide empirical evidence that GLDPC codes, when decoded with the APP decoder and equipped with the right fraction of GC nodes, can achieve substantial performance improvements compared to low-density parity-check (LDPC) codes.

Figures (10)

• Figure 1: The Tanner graph of GLDPC code ensembles.
• Figure 2: The thresholds $\epsilon^{\ast}$ in the BEC as a function of $t$ for both the $(\mathcal{C}_1, 2, 6, t)$ and $(\mathcal{C}_2, 2, 7, t)$ GLDPC ensembles.
• Figure 3: In (a), we show the design rates of $(\mathcal{C}_1,2,6,t)$ ensemble and $(\mathcal{C}_2,2,7,t)$ ensemble as functions of the threshold under the APP decoder on the BEC, and compare these rates with the channel capacity. In (b), we show the gaps to the channel capacity of these ensembles as functions of the threshold.
• Figure 4: The densities at the GC nodes corresponding to subcode $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively, which were obtained through Monte Carlo methods. The input messages of the GC nodes follow a Gaussian distribution with a mean of 3 and a variance of 6. It can be observed that the densities at the GC nodes can be well approximated by Gaussian distributions.
• Figure 5: The densities of messages sent by variable nodes in the $(\mathcal{C}_1,2,6,0.5)$ ensemble and $(\mathcal{C}_2,2,7,0.5)$ ensemble in the 5-th iteration on the BI-AWGN channel with SNR 4 were obtained through density evolution, as described in equation (\ref{['eq_de']}). In this process, $\Phi_G^C(P_l)$ was determined using Monte Carlo methods. It is evident that these densities deviate significantly from Gaussian distributions; however, they can be effectively approximated by Gaussian mixture distributions.

Theorems & Definitions (10)

• Definition 1: $(\mathcal{C}, J, K, t)$ GLDPC ensemble
• Definition 2: Extended Symmetry Conditions
• Lemma 1: Conditional Independence of Error Probability Under Symmetry
• Lemma 2
• Lemma 3
• Lemma 4
• Definition 3: message-invariant subcode
• Theorem 5
• Theorem 6
• Lemma 7