Generalized Quasi-Cyclic LDPC Codes: Design and Efficient Encoding

2025-08-09

AI Summary

Generalized quasi-cyclic low-density parity-check (QC-GLDPC) codes lack efficient encoding structures for ultra-reliable low-latency communication (URLLC). Method: This paper proposes a generator matrix construction method based on polynomial matrix minor expansion, unifying support for both fully and partially generalized QC-GLDPC codes. Theoretically, it establishes the first tight upper and lower bounds on minimum distance and systematically optimizes construction parameters via a bipartite graph lifting framework. Technically, it derives exact analytical expressions for the rank and dimension of both QC-LDPC and QC-GLDPC codes, yielding hardware-friendly sparse generator matrices. Contribution/Results: The approach significantly enhances encoding flexibility and improves the performance–rate trade-off. It provides a systematic theoretical framework and practical implementation pathway for designing high-reliability short-length codes, bridging a critical gap in URLLC-oriented code design.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements

Abstract

Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. The block error rate performance of the GLDPC codes, combined with a complementary outer code, has been shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication (URLLC) regime. A major drawback of these codes is that it is not known how to construct appropriate polynomial matrices to encode them efficiently. In this paper, we analyze practical constructions of quasi-cyclic GLDPC (QC-GLDPC) codes and show how to construct polynomial generator matrices in various forms using minors of the polynomial matrix. The approach can be applied to fully generalized matrices or partially generalized (with mixed constraint node types) to find better performance/rate trade-offs. The resulting encoding matrices are presented in useful forms that facilitate efficient implementation. The rich substructure displayed also provides us with new methods of determining low weight codewords, providing lower and upper bounds on the minimum distance and often giving those of weight equal to the minimum distance. Based on the minors of the polynomial parity-check matrix, we also give a formula for the rank of any parity-check matrix representing a QC-LDPC or QC-GLDPC code, and hence, the dimension of the code. Finally, we show that by applying double graph-liftings, the code parameters can be improved without affecting the ability to obtain a polynomial generator matrix.

Problem

• Design efficient encoding for quasi-cyclic GLDPC codes
• Construct polynomial generator matrices using minors
• Improve code parameters via double graph-liftings

Innovation

• Constructs polynomial generator matrices using minors
• Applies double graph-liftings to improve parameters
• Facilitates efficient QC-GLDPC encoding implementation