Counting and Entropy Bounds for Structure-Avoiding Spatially-Coupled LDPC Constructions

2026-01-14

AI Summary

This work addresses the design of large-scale spatially coupled low-density parity-check (SC-LDPC) codes with low error floors by eliminating harmful short cycles—particularly 4-cycles—introduced through edge-spreading and lifting operations. Leveraging the Clique Lovász Local Lemma (CLLL) and a Moser–Tardos-type constructive framework, the paper establishes, for the first time, an explicit lower bound on the number of structural configurations that avoid such detrimental substructures. By integrating Rényi entropy analysis, it further quantifies the diversity of valid solutions. A closed-form, parameterized lower bound on the number of 4-cycle-free partitioning matrices is derived via quantitative CLLL, and the count of non-isomorphic solutions under row and column permutations is characterized. These results provide a theoretical foundation for selecting coupling memory and lifting parameters while enabling a computable estimate of the feasible design space size.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• 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

Designing large coupling memory quasi-cyclic spatially-coupled LDPC (QC-SC-LDPC) codes with low error floors requires eliminating specific harmful substructures (e.g., short cycles) induced by edge spreading and lifting. Building on our work~\cite{r15} that introduced a Clique Lov\'asz Local Lemma (CLLL)-based design principle and a Moser--Tardos (MT)-type constructive approach, this work quantifies the size and structure of the feasible design space. Using the quantitative CLLL, we derive explicit lower bounds on the number of feasible edge-spreading and lifting assignments satisfying a given family of structure-avoidance constraints, and further obtain bounds on the number of non-equivalent solutions under row/column permutations. Moreover, via R\'enyi entropy bounds for the MT distribution, we provide a computable lower bound on the number of distinct solutions that the MT algorithm can output, giving a concrete diversity guarantee for randomized constructions. Specializations for eliminating 4-cycles yield closed-form bounds as functions of system parameters, offering a principled way to select the memory and lifting degree and to estimate the remaining search space.

Problem

• spatially-coupled LDPC
• short cycles
• structure-avoidance
• error floor
• substructures

Innovation

• spatially-coupled LDPC
• structure avoidance
• Lovász Local Lemma
• Moser–Tardos algorithm
• entropy bounds