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IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
On minimal tree realizations of linear codes
IEEE Transactions on Information Theory
Notes on Cutset Conditioning on Factor Graphs with Cycles
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ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
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IEEE Transactions on Information Theory
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IEEE Transactions on Information Theory
Analysis of absorbing sets and fully absorbing sets of array-based LDPC codes
IEEE Transactions on Information Theory
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IEEE Transactions on Information Theory
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On the application of BP decoding to convolutional and turbo codes
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
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Approximate Inference on Planar Graphs using Loop Calculus and Belief Propagation
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IEEE Transactions on Information Theory
Relaxation dynamics in stochastic iterative decoders
IEEE Transactions on Signal Processing
Dichotomy for Holant problems of Boolean domain
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Joint timing recovery and decoding algorithms for non-binary LDPC coded systems
Digital Signal Processing
A complete dichotomy rises from the capture of vanishing signatures: extended abstract
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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A generalized state realization of the Wiberg (1996) type is called normal if symbol variables have degree 1 and state variables have degree 2. A natural graphical model of such a realization has leaf edges representing symbols, ordinary edges representing states, and vertices representing local constraints. Such a graph can be decoded by any version of the sum-product algorithm. Any state realization of a code can be put into normal form without essential change in the corresponding graph or in its decoding complexity. Group or linear codes are generated by group or linear state realizations. On a cycle-free graph, there exists a well-defined minimal canonical realization, and the sum-product algorithm is exact. However, the cut-set bound shows that graphs with cycles may have a superior performance-complexity tradeoff, although the sum-product algorithm is then inexact and iterative, and minimal realizations are not well-defined. Efficient cyclic and cycle-free realizations of Reed-Muller (RM) codes are given as examples. The dual of a normal group realization, appropriately defined, generates the dual group code. The dual realization has the same graph topology as the primal realization, replaces symbol and state variables by their character groups, and replaces primal local constraints by their duals. This fundamental result has many applications, including to dual state spaces, dual minimal trellises, duals to Tanner (1981) graphs, dual input/output (I/O) systems, and dual kernel and image representations. Finally a group code may be decoded using the dual graph, with appropriate Fourier transforms of the inputs and outputs; this can simplify decoding of high-rate codes