Design of Root Protograph LDPC Codes Simultaneously Achieving Full Diversity and High Coding Gain

This paper presents a novel design framework for protograph-based LDPC codes that simultaneously achieves full diversity in block-fading channels (BFCs) and nearcapacity performance in additive white Gaussian noise channels (AWGNCs). By leveraging a Boolean approximation-based analysis–Diversity Evolution (DivE)–we derive structural constraints for generalized rootchecks that guarantee full diversity. Based on these constraints, we propose a protograph template tailored for two-block BFCs. Furthermore, we employ a genetic algorithm guided by density evolution to optimize the protograph edges within this template for superior AWGNC performance. The resulting codes effectively bridge the gap between diversityoriented and capacity-oriented designs, exhibiting robust performance across both channel environments.

💡 Research Summary

This paper introduces a unified design methodology for protograph‑based low‑density parity‑check (LDPC) codes that simultaneously achieves full diversity on block‑fading channels (BFCs) and near‑capacity performance on additive white Gaussian noise channels (AWGNCs). The authors first develop a Boolean‑approximation tool called Diversity Evolution (DivE) that tracks the evolution of fading information through belief‑propagation (BP) decoding. By representing each fading block with a binary variable (1 if the channel gain exceeds a threshold, 0 otherwise) and modeling check‑node and variable‑node updates as logical AND/OR operations, DivE yields a Boolean “fading function” for every variable node after an arbitrary number of iterations. A variable node attains full diversity when its final Boolean function equals the product of all block variables, i.e., it depends on every fading block.

Using DivE, the authors derive structural constraints for a new concept termed a generalized rootcheck. Unlike the classic rootcheck, which guarantees full diversity after a single BP iteration, a generalized rootcheck is defined in terms of the messages exchanged in the previous iteration: if all neighboring variable nodes (except the target one) carry either the opposite block’s fading variable or a full‑diversity expression, the check node will output the opposite block’s variable in the next iteration, thereby propagating diversity over multiple iterations. Two propositions prove that for the two‑block case (M = 2) any variable node connected to a generalized rootcheck becomes fully diverse after the corresponding iteration, and that if every information node becomes a generalized root node at some iteration, the whole code achieves full diversity.

Guided by these results, the paper proposes a protograph template for rate‑½ codes (n = 32, k = 16) that enforces a specific block mapping and row‑weight constraints. The template fixes a subset of edges to 0 or 1 and leaves a designable set D of edges free. Theorem 1 shows that any protograph belonging to the family induced by this template attains full diversity on a two‑block BFC within at most n/4 = 8 BP iterations, because each iteration guarantees two new information nodes become generalized root nodes. This guarantees the diversity requirement by construction, leaving only the AWGNC performance to be optimized.

To improve the coding gain on AWGNCs, the authors employ a genetic algorithm (GA) that searches only within the constrained family H(T). The GA initializes a population by randomly assigning 0/1 values to the designable edges, evaluates each candidate’s decoding threshold using reciprocal‑channel‑approximation density evolution (RCA‑DE), and uses the threshold (lower is better) as the fitness function. Standard GA operators (selection, crossover, mutation) evolve the population toward lower thresholds. Because all candidates already satisfy the diversity constraints, the GA focuses exclusively on maximizing AWGNC performance.

Simulation results demonstrate that the optimized protographs outperform the 5G‑NR root‑LDPC benchmark (γ_th ≈ 0.313 dB) by 0.13–0.44 dB in decoding threshold and reduce the capacity gap by 0.12–0.91 dB. In block‑fading simulations, the proposed codes achieve the full‑diversity slope (diversity order = 2) and provide 1–2 dB SNR savings compared with conventional root‑LDPC and generalized root‑protograph codes. Moreover, the designs retain a quasi‑cyclic structure after lifting (e.g., lifting factor Z = 128), ensuring hardware‑friendly implementation.

In summary, the paper delivers a novel, analytically grounded framework that bridges the long‑standing gap between diversity‑optimal codes for non‑ergodic fading channels and capacity‑approaching codes for AWGNCs. By integrating DivE‑based analysis, generalized rootchecks, a rigorously constrained protograph template, and GA‑driven AWGNC optimization, the authors produce LDPC codes that are simultaneously full‑diversity on BFCs and near‑capacity on AWGNCs, making them strong candidates for next‑generation wireless standards that must operate reliably under both fading and high‑throughput requirements.