5G LDPC Codes as Root LDPC Codes via Diversity Alignment

TL;DR

The paper addresses nonergodic block-fading channels by introducing Diversity Evolution (DivE), a Boolean-function framework that tracks how BP decoding messages evolve across iterations to reveal VN diversity. It then proposes a greedy block-mapping strategy, diversity alignment, that assigns protograph VNs to fading blocks to ensure full diversity for information bits without altering the base graph, aided by generalized rootchecks that emerge during iterative decoding. Applied to standard 5G-NR base graphs BG1 and BG2, the method achieves full diversity at higher code rates than random mappings, yielding a steeper high-SNR slope and lower BLER under Rayleigh block fading. This demonstrates that conventional AWGN-optimized LDPC codes can operate as Root-LDPC codes in a broader sense, provided an optimized block mapping, with implications for practical, diversity-friendly designs in 5G and beyond.

Abstract

This paper studies the diversity of protographbased quasi-cyclic low-density parity-check (QC-LDPC) codes over nonergodic block-fading channels under iterative beliefpropagation decoding. We introduce diversity evolution (DivE), a Boolean-function-based analysis method that tracks how the fading dependence of belief-propagation messages evolves across decoding iterations. Under a Boolean approximation of block fading, DivE derives a Boolean fading function for each variable node (VN) output (i.e., the a-posteriori reliability after iterative decoding), from which the VN diversity order can be directly determined. Building on this insight, we develop a greedy blockmapping search that assigns protograph VNs to fading blocks so that all information VNs achieve full diversity, while including the minimum additional parity VNs when full diversity is infeasible at the nominal rate. Numerical results on the 5G New Radio LDPC codes show that the proposed search finds block mappings that guarantee full diversity for all information bits without modifying the base-graph structure, yielding a markedly steeper high-SNR slope and lower BLER than random mappings.

Figures (8)

• Figure 1: Base matrix of BG1 5G-NR LDPC codes: Black squares indicate the edges in the corresponding protograph.
• Figure 2:
• Figure 3:
• Figure 5:
• Figure 6:

Theorems & Definitions (2)

• Definition 1: Block mapping
• Definition 2: Generalized rootcheck