List-based Optimization of Proximal Decoding for LDPC Codes

TL;DR

This work analyzes proximal decoding of LDPC codes on AWGN channels and identifies final-iteration oscillations that cause frame errors despite correct early progress. It introduces an ML-in-the-list step that targets the most problematic bit positions by tracking oscillation height $Δ_i^{(h)}$ and generating a candidate list for targeted ML decoding. A convergence analysis and simulations on a $(3,6)$-regular LDPC code show gains up to about $1$ dB over conventional proximal decoding, with the improvement depending on code and $E_b/N_0$. The approach offers a practical, linear-time enhancement to optimization-based decoding, with potential impact on high-reliability and large-scale systems such as massive MIMO.

Abstract

In this paper, the proximal decoding algorithm is considered within the context of additive white Gaussian noise (AWGN) channels. An analysis of the convergence behavior of the algorithm shows that proximal decoding inherently enters an oscillating behavior of the estimate after a certain number of iterations. Due to this oscillation, frame errors arising during decoding can often be attributed to only a few remaining wrongly decoded bit positions. In this letter, an improvement of the proximal decoding algorithm is proposed by establishing an additional step, in which these erroneous positions are attempted to be corrected. We suggest an empirical rule with which the components most likely needing correction can be determined. Using this insight and performing a subsequent ``ML-in-the-list'' decoding, a gain of up to 1 dB is achieved compared to conventional proximal decoding, depending on the decoder parameters and the code.

Figures (5)

• Figure 1: FER, DFR, and BER for $\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$mackay. Parameters: $\gamma =0.05,\omega = 0.05, \eta = 1.5, K=200$.
• Figure 2: Gradients $\nabla L\left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}}\right)$ and $\nabla h \left( \tilde{\boldsymbol{x}} \right)$ for a repetition code with $n=2$. Shown for $\boldsymbol{y} = -0.50.8$.
• Figure 3: Visualization of component $\tilde{x}_1$ for a decoding operation for a (3,6) regular LDPC code with $n=204, k=102$mackay. Parameters: $\gamma = 0.05, \omega = 0.05, \eta = 1.5, E_b/N_0 = 6dB$.
• Figure 4: Probability that $P(\hat{c}_{i'} \ne c_{i'})$ for a (3,6) regular LDPC code with $n=204, k=102$mackay. Indices $i'$ are ordered as in eq. (\ref{['eq:def:i_prime']}). Parameters: $\gamma = 0.05, \omega = 0.05, \eta = 1.5, E_\text{b}/N_0 = 6dB$, $10^9$ codewords.
• Figure 5: FER (- - -) and BER (---) of proximal decoding proximal_paper and the improved algorithm for a $\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$mackay. Parameters: $\gamma=0.05, \omega=0.05, \eta=1.5, K=200, N=8$.