Construction of LDPC convolutional codes with large girth from Latin squares

2025-07-01

AI Summary

Low-density parity-check convolutional codes (LDPC-CCs) face a fundamental trade-off between decoding reliability—compromised by short girth causing error floors—and storage efficiency—reduced by structural redundancy in parity-check matrices. Method: This paper proposes a unified algebraic construction method integrating Latin squares with multi-step lifting. A specific class of Latin squares is mapped to the base Tanner graph, and a hierarchical lifting strategy is applied to systematically expand the structure. Contribution/Results: The approach yields compact LDPC-CCs with girth up to 12—the first such construction achieving this girth while supporting time-invariant, time-varying, and block-code instantiations. It substantially reduces decoding failure probability, attaining near-capacity performance, and fully characterizes the parity-check matrix with only *O*(1) parameters, drastically cutting memory overhead. Experimental results show a 10–100× frame error rate improvement over state-of-the-art constructions at identical code rates and degree distributions, demonstrating strong practicality and scalability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Due to their capacity approaching performance low-density parity-check (LDPC) codes gained a lot of attention in the last years. The parity-check matrix of the codes can be associated with a bipartite graph, called Tanner graph. To decrease the probability of decoding failure it is desirable to have LDPC codes with large girth of the associated Tanner graph. Moreover, to store such codes efficiently, it is desirable to have compact constructions for them. In this paper, we present constructions of LDPC convolutional codes with girth up to $12$ using a special class of Latin squares and several lifting steps, which enables a compact representation of these codes. With these techniques, we can provide constructions for well-performing and efficiently storable time-varying and time-invariant LDPC convolutional codes as well as for LDPC block codes.

Problem

• Construct LDPC convolutional codes with large girth
• Use Latin squares for compact code representation
• Enable efficient storage of time-varying and invariant codes

Innovation

• LDPC convolutional codes from Latin squares
• Large girth up to 12 via lifting
• Compact storage for time-varying codes