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

Inki Kim, Hyuntae Ahn, Yongjune Kim, Hee-Youl Kwak, Dae-Young Yun, Sang-Hyo Kim

TL;DR

The paper targets LDPC codes that simultaneously provide full diversity on block-fading channels with $M=2$ and strong coding gain on AWGN channels. It combines a generalized-rootcheck protograph template (guaranteed by Diversity Evolution) with a GA guided by RCA-DE to optimize AWGN performance inside the template. The result is a family of rate-$1/2$ protographs for $M=2$ that ensure full diversity while achieving improved decoding thresholds and smaller capacity gaps compared to baselines like root-protograph and 5G-NR codes. This framework bridges diversity-oriented and capacity-oriented LDPC design, with demonstrated gains in both BFC and AWGN environments and clear paths to extension to larger $M$ and higher rates.

Abstract

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.

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

TL;DR

The paper targets LDPC codes that simultaneously provide full diversity on block-fading channels with and strong coding gain on AWGN channels. It combines a generalized-rootcheck protograph template (guaranteed by Diversity Evolution) with a GA guided by RCA-DE to optimize AWGN performance inside the template. The result is a family of rate- protographs for that ensure full diversity while achieving improved decoding thresholds and smaller capacity gaps compared to baselines like root-protograph and 5G-NR codes. This framework bridges diversity-oriented and capacity-oriented LDPC design, with demonstrated gains in both BFC and AWGN environments and clear paths to extension to larger and higher rates.

Abstract

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.
Paper Structure (13 sections, 3 theorems, 6 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 3 theorems, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Low-Density Parity-Check Codes
  4. Protograph Ensembles and QC-LDPC Codes
  5. Diversity Analysis of LDPC Codes under BP decoding in Block-Fading Channels
  6. BFC Model and Coded Diversity
  7. Diversity Analysis of LDPC Codes under BP Decoding
  8. Protograph Design Framework for Block-Fading and AWGN Channels
  9. Generalized Rootcheck Structure
  10. Family of Protographs Achieving Full Diversity
  11. Genetic Algorithm-Based Protograph Optimization
  12. Numerical Results
  13. Conclusion

Key Result

Proposition 1

For the BFC with $M=2$, if $v_i$ is connected to a generalized rootcheck at iteration $\ell$ and is the root node, then $v_i$ attains full diversity after iteration $\ell$.

Figures (6)

  • Figure 1: Boolean CN/VN updates in DivE (symbolic operations).
  • Figure 2: Comparison of rootcheck and generalized rootcheck structures.
  • Figure 3: Proposed protograph template for the BFC with $M=2$ ($n=32$, $k=16$) with full diversity ($d_c=2$). Entries marked by $b\in\{0,1\}$ are designable.
  • Figure 4: Flowchart of GA-based protograph optimization for AWGNCs.
  • Figure 5: BLER over the $M=2$ BFC for the 5G-NR LDPC, root-protograph LDPC, and the proposed code ($N=7744$, $R=1/2$). PEG-based lifting uses $Z=484$ for the root-protograph code and $Z=176$ for the 5G-NR and proposed codes.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 1: Block mapping
  • Definition 2: Boolean fading function
  • Definition 3: VN diversity
  • Definition 4: Full diversity
  • Definition 5: Rootcheck Boutros10
  • Remark 1
  • Definition 6: Generalized rootcheck
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 5 more