Probabilistic reasoning in intelligent systems: networks of plausible inference
Probabilistic reasoning in intelligent systems: networks of plausible inference
Introduction to the Theory of Computation
Introduction to the Theory of Computation
The size of bipartite graphs with a given girth
Journal of Combinatorial Theory Series B
The intractability of computing the minimum distance of a code
IEEE Transactions on Information Theory
Good error-correcting codes based on very sparse matrices
IEEE Transactions on Information Theory
Which codes have cycle-free Tanner graphs?
IEEE Transactions on Information Theory
Factor graphs and the sum-product algorithm
IEEE Transactions on Information Theory
The capacity of low-density parity-check codes under message-passing decoding
IEEE Transactions on Information Theory
On ensembles of low-density parity-check codes: asymptotic distance distributions
IEEE Transactions on Information Theory
Finite-length analysis of low-density parity-check codes on the binary erasure channel
IEEE Transactions on Information Theory
Variations on the Gallager bounds, connections, and applications
IEEE Transactions on Information Theory
Hardness of approximating the minimum distance of a linear code
IEEE Transactions on Information Theory
Distance distributions in ensembles of irregular low-density parity-check codes
IEEE Transactions on Information Theory
Quasicyclic low-density parity-check codes from circulant permutation matrices
IEEE Transactions on Information Theory
LDPC block and convolutional codes based on circulant matrices
IEEE Transactions on Information Theory
Regular and irregular progressive edge-growth tanner graphs
IEEE Transactions on Information Theory
Stopping set distribution of LDPC code ensembles
IEEE Transactions on Information Theory
Density evolution for asymmetric memoryless channels
IEEE Transactions on Information Theory
On the stopping distance and the stopping redundancy of codes
IEEE Transactions on Information Theory
Tightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes
IEEE Transactions on Information Theory
An efficient algorithm to find all small-size stopping sets of low-density parity-check matrices
IEEE Transactions on Information Theory
Constructing short-length irregular LDPC codes with low error floor
IEEE Transactions on Communications
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It is proven in this work that it is NP-complete to exhaustively enumerate small error-prone substructures in arbitrary, finite-length low-density parity-check (LDPC) codes. Two error-prone patterns of interest include stopping sets for binary erasure channels (BECs) and trapping sets for general memoryless symmetric channels. Despite the provable hardness of the problem, this work provides an exhaustive enumeration algorithm that is computationally affordable when applied to codes of practical short lengths n ≅ 500. By exploiting the sparse connectivity of LDPC codes, the stopping sets of size ≤ 13 and the trapping sets of size ≤ 11 can be exhaustively enumerated. The central theorem behind the proposed algorithm is a new provably tight upper bound on the error rates of iterative decoding over BECs. Based on a tree-pruning technique, this upper bound can be iteratively sharpened until its asymptotic order equals that of the error floor. This feature distinguishes the proposed algorithm from existing non-exhaustive ones that correspond to finding lower bounds of the error floor. The upper bound also provides a worst case performance guarantee that is crucial to optimizing LDPC codes when the target error rate is beyond the reach of Monte Carlo simulation. Numerical experiments on both randomly and algebraically constructed LDPC codes demonstrate the efficiency of the search algorithm and its significant value for finite-length code optimization.