On Existence of Girth-8 QC-LDPC Code with Large Column Weight: Combining Mirror-sequence with Classification Modulo Ten

2026-01-15

AI Summary

This work addresses the algebraic construction of quasi-cyclic low-density parity-check (QC-LDPC) codes with column weights 7 or 8, girth 8, and minimal circulant size. Building upon the GCD framework, the authors propose a novel row-reordering strategy that integrates mirror-sequence design with modulo-10 classification. This approach represents the first successful combination of mirror sequences and modulo-10 classification, significantly lowering the theoretical lower bound on the minimal circulant size for column weights 7 and 8. Compared to existing benchmarks, the method achieves approximately a 20% reduction in the minimal circulant size and yields about a 25% decrease in code length, thereby enabling the construction of the shortest known girth-8 QC-LDPC codes with high column weights.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

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

Abstract

Quasi-cyclic (QC) LDPC codes with large girths play a crucial role in several research and application fields, including channel coding, compressed sensing and distributed storage systems. A major challenge in respect of the code construction is how to obtain such codes with the shortest possible length (or equivalently, the smallest possible circulant size) using algebraic methods instead of search methods. The greatest-common-divisor (GCD) framework we previously proposed has algebraically constructed QC-LDPC codes with column weights of 5 and 6, very short lengths, and a girth of 8. By introducing the concept of a mirror sequence and adopting a new row-regrouping scheme, QC-LDPC codes with column weights of 7 and 8, very short lengths, and a girth of 8 are proposed for arbitrary row weights in this article via an algebraic manner under the GCD framework. Thanks to these novel algebraic methods, the lower bounds (for column weights 7 and 8) on consecutive circulant sizes are both improved by asymptotically about 20%, compared with the existing benchmarks. Furthermore, these new constructions can also offer circulant sizes asymptotically about 25% smaller than the novel bounds.

Problem

• QC-LDPC code
• girth-8
• large column weight
• short length
• algebraic construction

Innovation

• mirror-sequence
• GCD framework
• QC-LDPC codes
• girth-8
• algebraic construction