Good Codes Based on Very Sparse Matrices
Proceedings of the 5th IMA Conference on Cryptography and Coding
Finding all small error-prone substructures in LDPC codes
IEEE Transactions on Information Theory
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
Hardness of approximation results for the problem of finding the stopping distance in tanner graphs
FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
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
Efficient erasure correcting codes
IEEE Transactions on Information Theory
The capacity of low-density parity-check codes under message-passing decoding
IEEE Transactions on Information Theory
Low-density parity-check codes based on finite geometries: a rediscovery and new results
IEEE Transactions on Information Theory
Finite-length analysis of low-density parity-check codes on the binary erasure channel
IEEE Transactions on Information Theory
Hardness of approximating the minimum distance of a linear code
IEEE Transactions on Information Theory
On decoding of low-density parity-check codes over the binary erasure channel
IEEE Transactions on Information Theory
Regular and irregular progressive edge-growth tanner graphs
IEEE Transactions on Information Theory
On the stopping distance and the stopping redundancy of codes
IEEE Transactions on Information Theory
On the Stopping Redundancy of Reed–Muller Codes
IEEE Transactions on Information Theory
Improved Upper Bounds on Stopping Redundancy
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Computing the Stopping Distance of a Tanner Graph Is NP-Hard
IEEE Transactions on Information Theory
Turbo Decoding on the Binary Erasure Channel: Finite-Length Analysis and Turbo Stopping Sets
IEEE Transactions on Information Theory
Iterative reliability-based decoding of low-density parity check codes
IEEE Journal on Selected Areas in Communications
Parameterized Complexity
IEEE Transactions on Information Theory
On Linear Programming Decoding on a Quantized Additive White Gaussian Noise Channel
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
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In this work, we introduce an efficient algorithm to find all stopping sets, of size less than some threshold, of a fixed low-density parity-check (LDPC) matrix. The solution is inspired by the algorithm proposed by Rosnes and Ytrehus in 2005 to find an exhaustive list of all small-size turbo stopping sets in a turbo code. The efficiency of the proposed algorithm is demonstrated by several numerical examples. For instance, we have applied the algorithm to the well-known (3, 5)-regular (155, 64) Tanner code and found all stopping sets of size at most 18 in about 1 min on a standard desktop computer. Also, we have verified that the minimum stopping set size of the (4896, 2474) Ramanujan-Margulis code is indeed 24, and that the corresponding multiplicity is exactly 204. Furthermore, we have applied the algorithm to the IEEE 802.16e LDPC codes and determined the minimum stopping set size and the corresponding multiplicity exactly for these codes. Finally, as an application, we present a greedy algorithm to find a small number of redundant parity checks to add to the original parity-check matrix in order to remove all stopping sets in the corresponding Tanner graph of size less than the minimum distance. An extensive case study of the (155, 64) Tanner code illustrates the usefulness of the algorithm, and we present a 110 × 155 redundant parity-check matrix for this code with no stopping sets of size less than the minimum distance. Simulation results of iterative decoding on the binary erasure channel show performance improvements for low-to-medium erasure probabilities when this redundant parity-check matrix is used for decoding.