Finding Leafless Elementary Trapping Sets and Elementary Absorbing Sets of LDPC Codes is Hard

IEEE Trans. Inform. Theory

2019

Self Cite

The performance of low-density parity-check (LDPC) codes in the error floor region is closely related to some substructures of the code's Tanner graph, collectively referred to as trapping sets (TSs). In this paper, we study the asymptotic average number of different types of trapping sets such as elementary TSs (ETS), leafless ETSs (LETS), absorbing sets (ABS), elementary ABSs (EABS), and stopping sets (SS), in random variable-regular and irregular LDPC code ensembles. We demonstrate that, regardless of the type of the TS, as the code's length tends to infinity, the average number of a given structure tends to infinity, to a positive constant, or to zero, if the structure contains no cycle, only one cycle, or more than one cycle, respectively. For the case where the structure contains a single cycle, we derive the asymptotic expected multiplicity of the structure by counting the average number of its constituent cycles and all the possible ways that the structure can be constructed from the cycle. This, in general, involves computing the expected number of cycles of a certain length with a certain given combination of node degrees, or computing the expected number of cycles of a certain length expanded to the desired structure by the connection of trees to its nodes. The asymptotic results obtained in this work, which are independent of the block length and only depend on the code's degree distributions, are shown to be accurate even for finite-length codes.

Section: Introductionmentioning

confidence: 86%

“…The number of edges in the induced subgraph connected to unsatisfied check nodes and d v = 3, based of Corollary 2, the only non-empty classes (asymptotically) are those with b/a = d v − 2, whose members are simple cycles of length 2a. The average multiplicity of such classes can then be approximated by (9) or (10).…”

Section: Absorbing Sets (Abs)mentioning

confidence: 99%

Asymptotic Average Multiplicity of Structures Within Different Categories of Trapping Sets, Absorbing Sets, and Stopping Sets in Random Regular and Irregular LDPC Code Ensembles

Dehghan

IEEE Trans. Inform. Theory

2019

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“…There are close relationships between absorbing sets and TSs. In particular, elementary absorbing sets, which are a category of absorbing sets widely studied in the literature [8], [14], [15], [16], are in general a subset of leafless ETSs (LETSs) [11], [17], considered in this work.…”

mentioning

confidence: 99%

“…Within ETS category, leafless ETSs (LETS), those in which each variable node is connected to at least two satisfied (even-degree) check nodes, are the most harmful [11], [17]. Recently, Hashemi and Banihashemi [11] proposed a hierarchical graph-based characterization of LETS structures, dubbed dpl, for variable-regular LDPC codes.…”

mentioning

confidence: 99%

Construction of QC LDPC Codes With Low Error Floor by Efficient Systematic Search and Elimination of Trapping Sets

Karimi

2020

IEEE Trans. Commun.

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We propose a systematic design of protograph-based quasi-cyclic (QC) low-density parity-check (LDPC) codes with low error floor. We first characterize the trapping sets of such codes and demonstrate that the QC structure of the code eliminates some of the trapping set structures that can exist in a code with the same degree distribution and girth but lacking the QC structure. Using this characterization, our design aims at eliminating a targeted collection of trapping sets. Considering the parent/child relationship between the trapping sets in the collection, we search for and eliminate those trapping sets that are in the collection but are not a child of any other trapping set in the collection. An efficient layered algorithm is designed for the search of these targeted trapping sets. Compared to the existing codes in the literature, the designed codes are superior in the sense that they are free of the same collection of trapping sets while having a smaller block length, or a larger collection of trapping sets while having the same block length. In addition, the efficiency of the search algorithm makes it possible to design codes with larger degrees which are free of trapping sets within larger ranges compared to the state-of-the-art.

“…There are close relationships between absorbing sets and TSs. In particular, elementary absorbing sets, which are a category of absorbing sets widely studied in the literature [22,[41][42][43], are in general a subset of leafless ETSs (LETSs) [4,44], considered in this work. (The two categories of LETSs and elementary absorbing sets are identical for LDPC codes with maximum variable node degree 3.)…”

Section: Related Workmentioning

confidence: 99%

Construction of QC-LDPC Codes with Low Error Floor by Efficient Systematic Search and Elimination of Trapping Sets

Karimi

In this thesis, we propose a systematic design of binary protograph-based quasi-cyclic (QC) low-density parity-check (LDPC) codes with low error floor. We first characterize the trapping sets of such codes and demonstrate, using edge coloring techniques, that the QC structure of the code eliminates some of the trapping set structures that can exist in a code with the same degree distribution and girth but lacking the QC structure. Based on this characterization, our design aims at eliminating a targeted collection of trapping sets. Considering the parent/child relationship between the structures of the collection, we search for and eliminate those trapping sets that are in the collection but are not a child of any other trapping set in the collection. An efficient layered algorithm is designed for the search of these targeted trapping sets. Compared to the existing codes in the literature, the designed codes are superior in the sense that they are free of the same collection of trapping sets while having a smaller block length, or a larger collection of trapping sets while having the same block length. In addition, the efficiency of the search algorithm makes it possible to design codes with larger degrees which are free of trapping sets within larger ranges compared to the state-of-the-art. Moreover, we design finite-length binary irregular protograph-based QC-LDPC codes with good waterfall performance and low error floor. To achieve a low error floor, we eliminate a targeted set of dominant elementary trapping sets (ETS) L in the Tanner graph of the code. This is a more challenging task since the variety of problematic structures is much larger than that of regular codes. For a given rate and girth, the codes are designed to be free of the largest set of problematic ETSs for a given block length, or to have the shortest block length while a given set of ETSs is avoided. The design is based on a search algorithm that identifies whether any instance of any structure within L exists in the Tanner graph of the constructed code or not. The search algorithm performs this task with minimal complexity, making it feasible to construct practical codes by running the search algorithm a large number iii