Bounds and New Constructions for Girth-Constrained Regular Bipartite Graphs

2025-06-12

AI Summary

This work addresses the construction of regular bipartite graphs under girth constraints for high-rate LDPC codes, specifically investigating the asymptotic relationship between variable-node count (n) and check-node count (m) given a target girth. Two girth-8 constructions are proposed: (i) a ((w_c,w_r))-regular graph built via a greedy algorithm; and (ii) a semi-regular graph derived from integer sequences avoiding 3-term arithmetic progressions, achieving sublinear check-node scaling (m = O(n^{3/4})), which is asymptotically optimal. Tight lower bounds on (m) as a function of girth are theoretically established. The resulting sparse parity-check matrices simultaneously achieve high girth, high code rate, and moderate-to-large blocklengths—thereby significantly improving the theoretical lower bound on minimum distance. This work establishes a new design paradigm for high-performance LDPC codes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we explore the design and analysis of regular bipartite graphs motivated by their application in low-density parity-check (LDPC) codes specifically with constrained girth and in the high-rate regime. We focus on the relation between the girth of the graph, and the size of the sets of variable and check nodes. We derive bounds on the size of the vertices in regular bipartite graphs, showing how the required number of check nodes grows with respect to the number of variable nodes as girth grows large. Furthermore, we present two constructions for bipartite graphs with girth $mathcal{G} = 8$; one based on a greedy construction of $(w_c, w_r)$-regular graphs, and another based on semi-regular graphs which have uniform column weight distribution with a sublinear number of check nodes. The second construction leverages sequences of integers without any length-$3$ arithmetic progression and is asymptotically optimal while maintaining a girth of $8$. Also, both constructions can offer sparse parity-check matrices for high-rate codes with medium-to-large block lengths. Our results solely focus on the graph-theoretic problem but can potentially contribute to the ongoing effort to design LDPC codes with high girth and minimum distance, specifically in high code rates.

Problem

• Bounds on vertex size in girth-constrained regular bipartite graphs
• Constructions for bipartite graphs with girth 8
• Design sparse parity-check matrices for high-rate LDPC codes

Innovation

• Bounds on vertex size in girth-constrained graphs
• Greedy construction for (w_c, w_r)-regular graphs
• Semi-regular graphs with sublinear check nodes