Regular {4, 8} LDPC Codes and Their Lowerror Floors

Cole, Chad A.; Wilson, Stephen G.; Hall, E.K.; Giallorenzi, T.R.

doi:10.1109/milcom.2006.302365

Cited by 4 publications

(3 citation statements)

References 9 publications

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“…This behavior is well-established through density evolution, particularly in more precious terms of covariance evolution(Ornstein-Uhlenbeck stochastic process, [45,46]), observed in the waterfall region. Additionally, importance sampling is utilized in the error floor region for codes on the graph for phase transaction when the temperature falls below the critical point [47,48,49 Presented herein is a detailed exposition that offers evidence supporting the claim that the ground state solutions of the Ising Model are accurately represented by the parity-check matrix of QC-LDPC codes within the realm of (Hyperbolic) Toric Topology.…”

Section: Exploring the Connection Between Information Geometry And Hy...mentioning

confidence: 89%

Generalization of floor lifting for QC-LDPC codes

Usatyuk

Egorov

2017

2017 IEEE East-West Design &Amp; Test Symposium (EWDTS)

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The paper introduces an application of Information Geometry for describing the ground states of Ising models (Markov Random Fields) using parity-check matrices of Cyclic, Quasi-Cyclic codes on Toric and Spherical topologies. This approach establishes a connection between machine learning and error-correcting coding, specifically with regards to automorphism and the size of the circulant of the quasi-cyclic code, which determines the number of energy minima (ground states) of the quantum system. The proposed approach has implications for developing new embedding methods based on Trapping sets (TS(a,0)-codewords and TS(a,b)-pseudocodewords) optimized error-correcting codes using Statistical Physics and Number Geometry. The paper also demonstrates a relationship between the k-state Ising model's (Potts model) ground condition and lattices from number geometry, offering insights into fundamental principles governing complex systems like quantum computation and quantum error-correction, computer vision and NLP. The proposed code based embedding can be applied as factorization to diverse datasets from various domains, suggesting its effectiveness and versatility for different types of data and applications, such as quantum and classical computer vision, social networks, and optimal control problem data. Moreover, the paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-theart DNN architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to specific types (Cage-graph, Repeat Accumulate) of block and convolutional LDPC codes. The results of this research hold significant implications for the classification of codes and their application to quantum atomic models of chemicals on the spherical topology from the periodic system of Mendeleev. The study demonstrates that quasi cyclic codes and those with a complex structure of automorphisms correspond to certain types of chemical elements, particularly mixed automorphism Shu-Lin-Fossorier QC-LDPC code representing Carbon. The Quantum Approximate Optimization Algorithm (QAOA) utilized in the Sherrington-Kirkpatrick Ising model can be considered analogous to back-propagation loss function landscape in training DNN. This similarity creates a comparable problem with Trapping sets pseudo-codeword, resembling the Belief propagation (BP) soft decoding process. Furthermore, the layer depth (p) in QAOA correlates to the number of decoding BP iterations in the Wiberg (covering) decoding tree. The research findings have far-reaching implications, including but not limited to the classification of codes, development of novel embedding methods, construction of DNNs, and modeling physical and chemical properties of materials based on code representations. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, qua...

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Section: Exploring the Connection Between Information Geometry And Hy...mentioning

confidence: 89%

Generalization of floor lifting for QC-LDPC codes

Usatyuk

Egorov

2017

2017 IEEE East-West Design &Amp; Test Symposium (EWDTS)

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“…В работах [1][2][3][4] Коул предложил метод поиска треппин-сетов в LDPCкоде, путем использования метода выборки по значимости -метода Монте-Карло со смещенной оценкой, провоцирующего отказ алгоритма декодирования в узлах, потенциально содержащихся в треппин-сетах. Его метод позволяет ускорить исследование эффективности LDPC-кодов средней длины в области больших SNR.…”

Section: материалы и методыunclassified

“…Для решения этой задачи в работах [1][2][3][4] Коул предложил метод, использующий выборку по значимости. Метод предполагает построение дерева поиска и перебор различных вариантов импульсов в символьных узлах дерева.…”

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Construction of LDPC Codes Using Cole's Modified Importance Sampling Method

Usatjuk,

Egorov

2023

Proceedings of the SWSU

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Purpose of research is a modification of Cole's importance sampling method for finding trapping sets in an LDPC code to speed up their search.Methods. Cole proposed a method for searching for trapping sets in an LDPC code by using a Monte Carlo method that causes the decoding algorithm to fail at nodes potentially contained in trapping sets. His method makes it possible to accelerate the study of the efficiency of medium-length LDPC codes in the region of large SNRs. The modified method uses the properties of automorphisms of Tanner graphs, which make it possible to exclude a significant number of symbolic nodes from the enumeration. The modified method also provides for an ordered enumeration over subgraphs containing cycles.Results. The proposed method made it possible to speed up the search for trapping sets in Mackay's PEG(1008, 504) LDPC code by 5027 times compared to the Velasquez-Subramani method, and 43 times faster compared to the original Cole method. In the case of (2640, 1320) LDPC Margulis code, the proposed method is 28 times faster than the Velasquez-Subramani quasi-cyclic method and 134 times faster than the original Cole method.Conclusion. The result of experimental studies showed the possibility of using the developed method to improve the connectivity spectrum, increase the code distance of QC-LDPC codes. This made it possible to reduce the probability of a bit error at the decoder output by orders of magnitude at high signal-to-noise ratios in the AWGN channel.

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