Topologically Driven Methods for Construction Of Multi-Edge Type (Multigraph with nodes puncturing) Quasi-Cyclic Low-density Parity-check Codes for Wireless Channel, WDM Long-Haul and Archival Holographic Memory

TL;DR

This thesis presents advanced topologically driven methods for constructing multi-edge type QC-LDPC codes, optimizing error correction, complexity, and hardware implementation for various communication and storage systems.

Contribution

It introduces novel algorithms and topological insights for QC-LDPC code construction, including state-of-the-art lifting, distance estimation, and performance improvement techniques.

Findings

Effective construction of WDM Long-Haul error-correction codes.

Fast algorithms for code distance estimation and error-floor analysis.

Application of topological methods to improve code performance and hardware efficiency.

Abstract

In this Phd thesis discusses modern methods for constructing MET QC-LDPC codes with a given error correction ("waterfall, error-floor") and complexity (parallelism level according circulant size plus scheduler orthogonality of checks) profiles: 1. weight enumerators optimization, protograph construction using Density Evolution, MI (P/Exit-chart) and it approximation: Gaussian Approximation, Reciprocal-channel approximation and etc; 2. Covariance evolution and it approximation; 3. Lifting methods for QC codes construction:PEG, Guest-and-Test, Hill-Climbing with girth, EMD, ACE optimization; 4. Upper and lower bounds on code distance estimation and its parallel implementation using CPU/GPU; 5. Brouwer-Zimmerman and Number Geometry code distance estimation methods; 6. Importance Sampling for error-floor estimation; 7. Length and rate adaption methods for QC codes based on cyclic group decomposition; 8. Methods for interaction screening which allow to improve performance (decorrelate variables) under BP and it's approximation. We proposed several state-of-the-art methods: Simulated Annealing lifting for MET QC-LDPC codes construction; fast EMD and code distance estimation; floor scale modular lifting for lenght adaption; fast finite-length covariance evolution rate penalty from threshold for code construction and it hardware friendly compression for fast decoder's LLRs unbiasing due SNR's estimation error. We found topology reason's of efficient of such methods using topology thickening (homotopy of continuous and discrete curvature) under matched metric space which allow to generalize this idea to a class of nonlinear codes for Signal Processing and Machine Learning. Using the proposed algorithms several generations of WDM Long-Haul error-correction codes were built. It was applied for "5G eMBB" 3GPP TS38.212 and other applications like Flash storage, Compressed sensing measurement matrix.