290 research outputs found
An Efficient Algorithm for Finding Dominant Trapping Sets of LDPC Codes
This paper presents an efficient algorithm for finding the dominant trapping
sets of a low-density parity-check (LDPC) code. The algorithm can be used to
estimate the error floor of LDPC codes or to be part of the apparatus to design
LDPC codes with low error floors. For regular codes, the algorithm is initiated
with a set of short cycles as the input. For irregular codes, in addition to
short cycles, variable nodes with low degree and cycles with low approximate
cycle extrinsic message degree (ACE) are also used as the initial inputs. The
initial inputs are then expanded recursively to dominant trapping sets of
increasing size. At the core of the algorithm lies the analysis of the
graphical structure of dominant trapping sets and the relationship of such
structures to short cycles, low-degree variable nodes and cycles with low ACE.
The algorithm is universal in the sense that it can be used for an arbitrary
graph and that it can be tailored to find other graphical objects, such as
absorbing sets and Zyablov-Pinsker (ZP) trapping sets, known to dominate the
performance of LDPC codes in the error floor region over different channels and
for different iterative decoding algorithms. Simulation results on several LDPC
codes demonstrate the accuracy and efficiency of the proposed algorithm. In
particular, the algorithm is significantly faster than the existing search
algorithms for dominant trapping sets
Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning
The paper introduces the application of information geometry to describe the
ground states of Ising models by utilizing parity-check matrices of cyclic and
quasi-cyclic codes on toric and spherical topologies. The approach establishes
a connection between machine learning and error-correcting coding. This
proposed approach has implications for the development of new embedding methods
based on trapping sets. Statistical physics and number geometry applied for
optimize error-correcting codes, leading to these embedding and sparse
factorization methods. The paper establishes a direct connection between DNN
architecture and error-correcting coding by demonstrating how state-of-the-art
architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range
arena can be equivalent to of block and convolutional LDPC codes (Cage-graph,
Repeat Accumulate). QC codes correspond to certain types of chemical elements,
with the carbon element being represented by the mixed automorphism
Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and
the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix
are elaborated upon in detail. The Quantum Approximate Optimization Algorithm
(QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous
to the back-propagation loss function landscape in training DNNs. This
similarity creates a comparable problem with TS pseudo-codeword, resembling the
belief propagation method. Additionally, the layer depth in QAOA correlates to
the number of decoding belief propagation iterations in the Wiberg decoding
tree. 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,
quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text
overlap with arXiv:2109.08184 by other author
Modern Coding Theory: The Statistical Mechanics and Computer Science Point of View
These are the notes for a set of lectures delivered by the two authors at the
Les Houches Summer School on `Complex Systems' in July 2006. They provide an
introduction to the basic concepts in modern (probabilistic) coding theory,
highlighting connections with statistical mechanics. We also stress common
concepts with other disciplines dealing with similar problems that can be
generically referred to as `large graphical models'.
While most of the lectures are devoted to the classical channel coding
problem over simple memoryless channels, we present a discussion of more
complex channel models. We conclude with an overview of the main open
challenges in the field.Comment: Lectures at Les Houches Summer School on `Complex Systems', July
2006, 44 pages, 25 ps figure
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