On the Performance of Low-complexity Decoders of LDPC Codes

TL;DR

This work derives a unified closed-form lower bound on the average BER of LDPC code ensembles under belief propagation decoding with a limited number of iterations $l$, by connecting performance to the expected minimum Hamming weight of root-constrained tree codes. For regular LDPC ensembles with fixed degrees $(J,K)$, the bound attains a double-exponential form $h 2^{-a 2^{b l}}$ with $b=\log_2(J-1)$ in the low-iteration regime, and it is shown to be tight relative to known upper bounds in that regime. The irregular ensemble analysis extends the framework with an iterative, graph-theoretic method to tighten the bound using distance-distribution considerations, offering practical means to assess decoding performance under resource constraints. Simulations corroborate the theoretical findings, illustrating how the bounds bracket the actual BER for small $l$ and guide decoding-resource planning for high-throughput, low-latency communication systems.

Abstract

Efficient decoding is crucial to high-throughput and power-sensitive wireless communication scenarios. A theoretical analysis of the performance-complexity tradeoff toward low-complexity decoding is required for a better understanding of the fundamental limits in the above-mentioned scenarios. This study aims to explore the performance of LDPC codes under belief propagation (BP) decoding with complexity constraints. In other words, for a small number of iterations, we present a closed-form lower bound on the bit error rate (BER) of LDPC codes as a function of complexity. Specifically, for the regular LDPC code ensembles, the dominant term in the order of the lower bound we provide matches that of the upper bound given by Lentmaier. Furthermore, for irregular LDPC code ensembles, in addition to adopting the approach used for regular codes, we also propose a method to iteratively obtain the lower bound on the average BER.

Figures (7)

• Figure 1: Parity-check matrix and Tanner graph for an LDPC code with a code length of $4$. In the Tanner graph, circles represent variable nodes, and squares represent check nodes. The $i$-th variable node receives channel message LLR$_i$
• Figure 2: A computation graph of height $4$ for $v_1$ in Fig. \ref{['fig1']}. Black nodes represent the first occurrence of a node, while white nodes indicate subsequent occurrences.
• Figure 3: An example codeword from $\mathcal{C}_4(v_1,G_1)$, where solid and hollow circles represent variable nodes assigned the values 1 and 0, respectively. Note that $T_{4}(v_1,G_1)$ contains repeated variable nodes; circles with the same color indicate the same variable node, which takes the same value (either 0 or 1) across all occurrences.
• Figure 4: $\gamma$ vs. $J$ for the bound of LDPC code ensembles, illustrating the tightness of the proposed lower bound.
• Figure 5: The computation graphs and the valid trees (in solid blue) contained within them. Here, the computation graph differs from the one in the body of the paper, as repeated nodes appear only once.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Remark
• Remark
• Theorem 2
• Lemma 1
• proof : Proof of Lemma \ref{['lemma_regular_w']}