An Efficient Instanton Search Algorithm for LP Decoding of LDPC Codes Over the BSC

We consider linear programming (LP) decoding of a fixed low-density parity-check (LDPC) code over the binary symmetric channel (BSC). The LP decoder fails when it outputs a pseudo-codeword which is not equal to the transmitted codeword. We design an efficient algorithm termed the Instanton Search Algorithm (ISA) which generates an error vector called the BSC-instanton. We prove that: (a) the LP decoder fails for any error pattern with support that is a superset of the support of an instanton; (b) for any input, the ISA outputs an instanton in the number of steps upper-bounded by twice the number of errors in the input error vector. We then find the number of unique instantons of different sizes for a given LDPC code by running the ISA sufficient number of times.This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

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

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.Comment: Phd Thesis, 176 pages, in Russian, 62 pictures, 13 tables, 5 appendix including links to binary and source code