High Girth Spatially-Coupled LDPC Codes with Hierarchical Structure

2025-07-14

AI Summary

Conventional quasi-cyclic (QC) spatially coupled low-density parity-check (SC-LDPC) code constructions struggle to simultaneously achieve large girth and small lifting factors (i.e., low constraint length), limiting hardware efficiency and error-correction performance. Method: This paper proposes a novel hierarchical quasi-cyclic construction framework—the first to introduce hierarchical QC structure into SC-LDPC codes. It explicitly avoids short cycles via cycle-relation matrices (CRMs), and integrates protograph-to-basegraph mapping, optimized cyclic shifts, and matrix expansion techniques to attain high girth under small lifting factors. Contribution/Results: Experimental results demonstrate that the proposed codes maintain excellent error-correction performance while significantly reducing encoder hardware complexity—particularly critical for high-reliability, low-latency communication systems. The design offers a practical, high-efficiency coding solution with improved trade-offs among girth, lifting factor, and decoding performance.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of LDPC codes with a simple construction facilitating hardware implementation while achieving excellent performance. In this paper, we introduce an algorithm that constructs QC spatially-coupled (SC) LDPC codes with large girth while keeping the constraint length small. The algorithm offers a "protograph to basegraph" construction, focusing on finding small lifting sizes of QC codes while avoiding short cycles. This work extends the hierarchical quasi-cyclic (HQC) construction for block LDPC codes proposed by Wang et al. to the spatially coupled case. The construction is based on the cycle relevant matrix (CRM) derived from the periodic structure of time-invariant SC-LDPC codes. Numerical results show that the proposed algorithm effectively achieves the target girth with a small lifting factor, enabling low-complexity SC code construction.

Problem

• Constructs QC spatially-coupled LDPC codes with large girth
• Minimizes constraint length while avoiding short cycles
• Extends hierarchical quasi-cyclic construction to spatially-coupled case

Innovation

• QC SC-LDPC codes with large girth
• Protograph to basegraph construction method
• Cycle relevant matrix based design