In this paper, we study the graphical structure of elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree d l and girth g, we identify all the non-isomorphic structures of an arbitrary class of (a, b) ETSs, where a is the number of variable nodes and b is the number of odd-degree check nodes in the induced subgraph of the ETS. Our study leads to a simple characterization of dominant classes of ETSs (those with relatively small values of a and b) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S in the class is a layered superset (LSS) of a short cycle, where the term "layered" is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for d l = 3, 4, 5, 6, g = 6, 8 and a, b ≤ 10 in this paper. The results of this paper can be used for the error floor analysis and for the design of LDPC codes with low error floors. I. INTRODUCTIONT HE performance of low-density parity-check (LDPC) codes under iterative decoding algorithms in the error floor region is closely related to the problematic structures of the code's Tanner graph [11], [25], [27], [32], [16], [26], [35], [36], [8], [5], [9], [15], [38]. Following the nomenclature of [27], here, we collectively refer to such structures as trapping sets. The most common approach for classifying the trapping sets is by a pair (a, b), where a is the size of the trapping set and b is the number of odd-degree (unsatisfied) check nodes in the subgraph induced by the set in the Tanner graph of the code. Among the trapping sets, the so-called elementary trapping sets (ETS) are known to be the main culprits [27], [16], [5], [26], [15], [38]. These are trapping sets whose induced subgraph only contains check nodes of degree one or two. For a given LDPC code, the knowledge of dominant trapping sets, i.e., those that are most harmful, is important. Such knowledge can be used to estimate the error floor [5], to modify the decoder to lower the error floor [4], [12], [22], or to design codes with low error floor [14], [1]. 2 While the knowledge of dominant trapping sets is most helpful in the design and analysis of LDPC codes, attaining such knowledge is generally a hard problem [24]. Much research has been devoted to devising efficient search algorithms for finding small (dominant) trapping sets, see, e.g., [33], [5], [28], [36], [34], [2], [22], [18], and to the (partial) characterization of such sets [7], [30] [10], [17], [13], [6]. Asymptotic analysis of trapping sets has also been carried out in [26], [3], [29], [20], [19], [10]. Laendner et al. [17] st...
This paper presents an efficient algorithm for finding the dominant trapping sets of a low-density parity-check (LDPC) code. The algorithm can be used to estimate the error floor of LDPC codes or to be used as a tool to design LDPC codes with low error floors. For regular codes, the algorithm is initiated with a set of short cycles as the input. For irregular codes, in addition to short cycles, variable nodes with low degree and cycles with low approximate cycle extrinsic message degree (ACE) are also used as the initial inputs. The initial inputs are then expanded recursively to dominant trapping sets of increasing size. At the core of the algorithm lies the analysis of the graphical structure of dominant trapping sets and the relationship of such structures to short cycles, low-degree variable nodes and cycles with low ACE. The algorithm is universal in the sense that it can be used for an arbitrary graph and that it can be tailored to find a variety of graphical objects, such as absorbing sets and Zyablov-Pinsker (ZP) trapping sets, known to dominate the performance of LDPC codes in the error floor region over different channels and for different iterative decoding algorithms. Simulation results on several LDPC codes demonstrate the accuracy and efficiency of the proposed algorithm. In particular, the algorithm is significantly faster than the existing search algorithms for dominant trapping sets.
Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle c of length ℓ(c) in the base graph such that the inverse image of c in the lifted graph consists of only cycles of length strictly larger than ℓ(c). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, encoder and decoder structure, and in some cases, the code rate. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.
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