On the Complexity of finding Stopping Distance in Tanner Graphs

Two decision problems related to the computation of stopping sets in Tanner graphs are shown to be NP-complete. NP-hardness of the problem of computing the stopping distance of a Tanner graph follows as a consequenceComment: A decision problem proved NP-complete in the earlier version was not equivalent to stopping distance problem for Tanner graphs. Now correcte

Upper Bounding the Performance of Arbitrary Finite LDPC Codes on Binary Erasure Channels

Assuming iterative decoding for binary erasure channels (BECs), a novel tree-based technique for upper bounding the bit error rates (BERs) of arbitrary, finite low-density parity-check (LDPC) codes is provided and the resulting bound can be evaluated for all operating erasure probabilities, including both the waterfall and the error floor regions. This upper bound can also be viewed as a narrowing search of stopping sets, which is an approach different from the stopping set enumeration used for lower bounding the error floor. When combined with optimal leaf-finding modules, this upper bound is guaranteed to be tight in terms of the asymptotic order. The Boolean framework proposed herein further admits a composite search for even tighter results. For comparison, a refinement of the algorithm is capable of exhausting all stopping sets of size <14 for irregular LDPC codes of length n=500, which requires approximately 1.67*10^25 trials if a brute force approach is taken. These experiments indicate that this upper bound can be used both as an analytical tool and as a deterministic worst-performance (error floor) guarantee, the latter of which is crucial to optimizing LDPC codes for extremely low BER applications, e.g., optical/satellite communications.Comment: To appear in the Proceedings of the 2006 IEEE International Symposium on Information Theory, Seattle, WA, July 9 - 14, 200

Exhausting Error-Prone Patterns in LDPC Codes

It is proved in this work that exhaustively determining bad patterns in arbitrary, finite low-density parity-check (LDPC) codes, including stopping sets for binary erasure channels (BECs) and trapping sets (also known as near-codewords) for general memoryless symmetric channels, is an NP-complete problem, and efficient algorithms are provided for 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 efficiently exhaustively determined for the first time, and the resulting exhaustive list is of great importance for code analysis and finite code optimization. The featured tree-based narrowing search distinguishes this algorithm from existing ones for which inexhaustive methods are employed. One important byproduct is a pair of upper bounds on the bit-error rate (BER) & frame-error rate (FER) iterative decoding performance of arbitrary codes over BECs that can be evaluated for any value of the erasure probability, including both the waterfall and the error floor regions. The tightness of these upper bounds and the exhaustion capability of the proposed algorithm are proved when combining an optimal leaf-finding module with the tree-based search. These upper bounds also provide a worst-case-performance guarantee which is crucial to optimizing LDPC codes for extremely low error rate applications, e.g., optical/satellite communications. Extensive numerical experiments are conducted that include both randomly and algebraically constructed LDPC codes, the results of which demonstrate the superior efficiency of the exhaustion algorithm and its significant value for finite length code optimization.Comment: submitted to IEEE Trans. Information Theor