Fast and Accurate Error Floor Estimation of Quantized Iterative Decoders for Variable-Regular LDPC Codes

2016 IEEE International Conference on Communications (ICC)

2016

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In this paper, we propose a new characterization for elementary trapping sets (ETSs) of variableregular low-density parity-check (LDPC) codes. Recently, Karimi and Banihashemi proposed a characterization of ETSs, which was based on viewing an ETS as a layered superset (LSS) of a short cycle in the code's Tanner graph. A notable advantage of LSS characterization is that it corresponds to a simple LSS-based search algorithm (expansion technique) that starts from short cycles of the graph and finds the ETSs with LSS structure efficiently. Compared to the LSS-based characterization of Karimi and Banihashemi, which is based on a single LSS expansion technique, the new characterization involves two additional expansion techniques. The introduction of the new techniques mitigates two problems that LSS-based characterization/search suffers from: (1) exhaustiveness: not every ETS structure is an LSS of a cycle, (2) search efficiency: LSS-based search algorithm often requires the enumeration of cycles with length much larger than the girth of the graph, where the multiplicity of such cycles increases rapidly with their length. We prove that using the three expansion techniques, any ETS structure can be obtained starting from a simple cycle, no matter how large the size of the structure a or the number of its unsatisfied check nodes b are, i.e., the characterization is exhaustive. We also demonstrate that for the proposed characterization/search to exhaustively cover all the ETS structures within the (a, b) classes with a ≤ a max , b ≤ b max , for any value of a max and b max , the length of the short cycles required to be enumerated is less than that of the LSS-based characterization/search. We, in fact, show that such a length for the proposed search algorithm is minimal. We also prove that the three expansion techniques, proposed here, are the only expansions needed for characterization of ETS structures starting from simple cycles in the graph, if one requires each and every intermediate sub-structure to be an ETS as well. Extensive simulation results are provided to show that, compared to LSS-based search, significant improvement in search speed and memory requirements can be achieved.In an earlier paper [13], we complemented the results of [17]. By careful examination of NA structures, we demonstrated that they are all LSSs of some basic graphical structures which are slightly more complex than cycles. These basic structures were called prime with respect to the LSS property, or "LSS-prime" in brief, implying that they are not layered super sets of smaller ETSs. Results of [17] (and its corrections in [14]) along with those of [13] provided a simple LSS-based search algorithm that can find all the instances of ETSs of variable-regular LDPC codes, for any range of a and b values, in a guaranteed fashion, starting from the instances of LSS-prime structures. The LSS-based search algorithm, however, suffers from a major drawback. Although, the search algorithm itself is simple, to find the dominant ETSs in a guaranteed fa...

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An efficient exhaustive search algorithm for elementary trapping sets of variable-regular LDPC codes

Hashemi

2016 IEEE International Conference on Communications (ICC)

2016

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“…For a given LDPC code, the knowledge of TSs, in general, and that of LETSs, in particular, is useful in estimating the error floor [10], [11], devising decoding algorithms [12], or designing codes [13], [14] with low error floors. In any such application, one would need to find a list of dominant TSs which are the main contributors to the error floor.…”

Section: Introductionmentioning

confidence: 99%

Hardness Results on Finding Leafless Elementary Trapping Sets and Elementary Absorbing Sets of LDPC Codes

Dehghan

IEEE Trans. Inform. Theory

2019

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Leafless elementary trapping sets (LETSs) are known to be the problematic structures in the error floor region of low-density parity-check (LDPC) codes over the additive white Gaussian (AWGN) channel under iterative decoding algorithms. While problems involving the general category of trapping sets, and the subcategory of elementary trapping sets (ETSs), have been shown to be NP-hard, similar results for LETSs, which are a subset of ETSs are not available. In this paper, we prove that, for a general LDPC code, finding a LETS of a given size a with minimum number of odd-degree check nodes b is NP-hard to approximate within any approximation factor. We also prove that finding the minimum size a of a LETS with a given b is NP-hard to approximate within any approximation factor. Similar results are proved for elementary absorbing sets, a popular subcategory of LETSs. Index Terms: Low-density parity-check (LDPC) codes, trapping sets (TS), elementary trapping sets (ETS), leafless elementary trapping sets (LETS), absorbing sets (ABS), elementary absorbing sets (EABS), computational complexity, NP-hardness.

“…For a given LDPC code, the knowledge of trapping sets is important. Such knowledge can be used, for example, to estimate the error floor [25], [31], to devise decoding algorithms with low error floor [37], [19], or to design codes with low error floor [14], [2], [3], [18], [24]. There are numerous works on the characterization and search algorithms for trapping sets [36], [34], [35], [1], [27], [6], [5], [20], [7], [29], [38], [16], [24], [19], [8], [15], [10], [12].…”

Section: Introductionmentioning

confidence: 99%

Characterization and efficient exhaustive search algorithm for elementary trapping sets of irregular LDPC codes

Hashemi

2017 IEEE International Symposium on Information Theory (ISIT)

2017

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In this paper, we propose a characterization of elementary trapping sets (ETSs) for irregular lowdensity parity-check (LDPC) codes. These sets are known to be the main culprits in the error floor region of such codes. The characterization of ETSs for irregular codes has been known to be a challenging problem due to the large variety of non-isomorphic ETS structures that can exist within the Tanner graph of these codes. This is a direct consequence of the variety of the degrees of the variable nodes that can participate in such structures. The proposed characterization is based on a hierarchical graphical representation of ETSs, starting from simple cycles of the graph, or from single variable nodes, and involves three simple expansion techniques: degree-one tree (dot), path and lollipop, thus, the terminology dpl characterization. A similar dpl characterization was proposed in an earlier work by the authors for the leafless ETSs (LETSs) of variable-regular LDPC codes. The present paper generalizes the prior work to codes with a variety of variable node degrees and to ETSs that are not leafless. The proposed dpl characterization corresponds to an efficient search algorithm that, for a given irregular LDPC code, can find all the instances of (a, b) ETSs with size a and with the number of unsatisfied check nodes b within any range of interest a ≤ a max and b ≤ b max , exhaustively. Although, (brute force) exhaustive search algorithms for ETSs of irregular LDPC codes exist, to the best of our knowledge, the proposed search algorithm is the first of its kind, in that, it is devised based on a characterization of ETSs that makes the search process efficient. Extensive simulation results are presented to show the versatility of the search algorithm, and to demonstrate that, compared to the literature, significant improvement in search speed can be obtained.