Generalization of Tanner's minimum distance bounds for LDPC codes

Shin, Min‐Ho; Kim, Joon-Sung; Song, Hong‐Yeop

doi:10.1109/lcomm.2005.03002

Cited by 6 publications

(1 citation statement)

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“…The spectral gap is closely associated with certain topological properties, such as the Bethe tree or lattice, Bethe permanent (pseudocodewords TS(a,b)) [146,147], and permanent (codewords TS(a,0)) [148,149]. Additionally, the graph spectral bound [144,145,151] is also related to these properties. Neural networks that have a minimal clustering coefficient tend to outperform others.…”

Section: Ising Models Ground States Configuration From Famous Qc-ldpc...mentioning

confidence: 99%

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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