Small Trapping Sets Research Articles

Low-density parity-check codes: Highway to channel capacity

Low-density parity-check (LDPC) codes are not only capacity-approaching, but also greatly suitable for high-throughput implementation. Thus, they are the most popular codes for high-speed data transmission in the past two decades. Thanks to the low-density property of their parity-check matrices, the optimal maximum a posteriori probability decoding of LDPC codes can be approximated by message-passing decoding with linear complexity and highly parallel nature. Then, it reveals that the approximation has to carry on Tanner graphs without short cycles and small trapping sets. Last, it demonstrates that well-designed LDPC codes with the aid of computer simulation and asymptotic analysis tools are able to approach the channel capacity. Moreover, quasi-cyclic (QC) structure is introduced to significantly facilitate their high-throughput implementation. In fact, compared to the other capacity-approaching codes, QC-LDPC codes can provide better area-efficiency and energy-efficiency. As a result, they are widely applied in numerous communication systems, e.g., Landsat satellites, Chang'e Chinese Lunar mission, 5G mobile communications and so on. What's more, its extension to non-binary Galois fields has been adopted as the channel coding scheme for BeiDou navigation satellite system.

QC-LDPC Codes With Large Column Weight and Free of Small Size ETSs

An approach to improve the performance of QC-LDPC codes is the removal of harmful trapping sets by increasing the girth. However, constructing these LDPC codes with large column weights and girth more than 8 is not easy. We are concerned with protograph-based LDPC codes with large column weights and free of small size trapping sets. We use the edge-coloring technique and some concepts from graph theory such as rainbow cycles to show that for large column weights the removal of all 8-cycles but the ones we call <inline-formula> <tex-math notation="LaTeX">$\textit {rainbow 8-cycle}$ </tex-math></inline-formula> causes the elimination of several small size trapping sets. We provide a detailed theoretical analysis of these harmful trapping sets. Then, we apply them to array-based LDPC codes to significantly simplify and optimize the necessary and sufficient conditions to eliminate those 8-cycles from the Tanner graph. The given exponent matrices and simulation results show the impact of this simplification and the removal of the above mentioned 8-cycles.

Construction of Time Invariant Spatially Coupled LDPC Codes Free of Small Trapping Sets

In this paper, we propose a design technique for the construction of variable-regular time-invariant spatially-coupled low-density parity-check (SC-LDPC) codes with small constraint length and low error floor. The proposed technique reduces the error floor by imposing simple constraints on the short cycles in the code's Tanner graph, which in turn, result in the elimination of the most dominant trapping sets of the code. In some cases, we also derive lower bounds on the syndrome former memory for satisfying such constraints. The designed codes are superior to the state-of-the-art in terms of error floor performance and/or decoding complexity and latency.

Layered Construction of Quasi-Cyclic LDPC Codes

A layered construction of LDPC codes on 3-dimensional rectangular lattices is explored in this letter. By analyzing the governing conditions of in-layer polygons and inter-layer polygons, and elaborately selecting the slopes of in-layer and inter-layer lines, we remove all the 6-cycles as well as inter-layer 8-cycles which results in a class of LDPC codes without small trapping sets. We further remove the in-layer 8-cycles to design a class of LDPC codes with girth 10.

QC-LDPC construction free of small size elementary trapping sets based on multiplicative subgroups of a finite field

Trapping sets significantly influence the performance of low-density parity-check codes. An $ (a, b) $ elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code, where $ a $ and $ b $ denote the size and the number of unsatisfied check-nodes in the ETS, respectively. The smallest size of an ETS in $ (3, n) $-regular LDPC codes with girth 6 is 4. In this paper, we provide sufficient conditions to construct fully connected $ (3, n) $-regular algebraic-based QC-LDPC codes with girth 6 whose Tanner graphs are free of $ (a, b) $ ETSs with $ a\leq5 $ and $ b\leq2 $. We apply these sufficient conditions to the exponent matrix of a new algebraic-based QC-LDPC code with girth at least 6. As a result, we obtain the maximum size of a submatrix of the exponent matrix which satisfies the sufficient conditions and yields a Tanner graph free of those ETSs with small size. Some algebraic-based QC-LDPC code constructions with girth 6 in the literature are special cases of our construction. Our experimental results show that removing ETSs with small size contribute to have better performance curves in the error floor region.

Construction of Girth-8 QC-LDPC Codes Free of Small Trapping Sets

In this letter, we propose a new technique to design quasi-cyclic low-density parity-check (QC-LDPC) codes with girth 8 and variable node degrees 3 and 4. Small trapping sets are avoided in the designed codes by imposing simple constraints on the cycles of length 8 in the code’s Tanner graph. We also derive lower bounds on the block length of the codes that can satisfy such constraints. Simulation results are provided to demonstrate the superiority of the designed codes in comparison with similar codes in the literature.

On the Construction of LDPC Codes Free of Small Trapping Sets by Controlling Cycles

Low-density parity-check (LDPC) codes exhibit excellent error correcting capability. However, small trapping sets in the Tanner graph are harmful to the iterative decoding algorithm. In this letter, we present a method of constructing $(3, n) $ girth-eight quasi-cyclic LDPC codes with low error floor by removing the small trapping sets from the Tanner graph. To address this issue, we analyze the relationship between eight-cycles and small trapping sets of Tanner graphs based on fully connected base graphs without parallel edges. We find that if some eight-cycles are not found in the Tanner graphs, any elementary trapping set in the range of $a \leq 8$ and $b \leq 3$ is removed naturally. We also derive a lower bound on the permutation size for the construction of such codes. The experimental simulation shows a favorable error rate performance with lower error floor over additive white Gaussian noise channels.

A new two‐stage decoding scheme with unreliable path search to lower the error‐floor for low‐density parity‐check codes

In this study, the authors propose a new two-stage decoding scheme for low-density parity-check codes to lower the error-floor, which consists of the conventional belief propagation (BP) decoding algorithm as the first-stage decoding and the re-decodings with manipulated log-likelihood ratios (LLRs) of variable nodes as the second-stage decoding. In the first-stage decoding, an early stopping criterion is proposed for early detection of decoding failure and the candidate set of the variable nodes are determined, which can be partly included in the small trapping sets. In the second-stage decoding, the scores of the variable nodes in the candidate set are computed by the proposed unreliable path search algorithm and the variable nodes are sorted in ascending order by their scores for the re-decoding trials. Each re-decoding trial is performed by BP decoding algorithm with manipulated LLR of a selected variable node in the candidate set one at a time with the second early stopping criterion. The parallel unreliable path search algorithm is also proposed for practical application of the proposed algorithm. Numerical results show that the proposed early stopping criteria and the proposed decoding algorithms for the second-stage decoding can correct most of the unsuccessfully decoded codewords by the first-stage decoding in the error-floor region.

Check‐hybrid GLDPC codes: Systematic elimination of trapping sets and guaranteed error correction capability

ABSTRACTIn this paper, we propose a new approach to construct a class of check‐hybrid generalized low‐density parity‐check (CH‐GLDPC) codes, which are free of small trapping sets. The approach is based on converting some selected check nodes involving a trapping set into super checks corresponding to a 2‐error‐correcting component code. Specifically, we follow 2 main purposes to construct the check‐hybrid codes; first, on the basis of the knowledge of trapping sets of an LDPC code, single parity checks are replaced by super checks to disable the trapping sets. We show that by converting specified single check nodes, denoted as critical checks, to super checks in a trapping set, the parallel bit flipping decoder corrects the errors on a trapping set. The second purpose is to minimize the rate loss through finding the minimum number of such critical checks. We also present an algorithm to find critical checks in a trapping set of a column‐weight 3 LDPC code of girth 8 and then provide upper bounds on the minimum number of such critical checks such that the decoder corrects all error patterns on elementary trapping sets. Guaranteed error correction capability of the CH‐GLDPC codes is also studied. We show that a CH‐GLDPC code in which each variable node is connected to 2 super checks corresponding to a 2‐error‐correcting component code corrects up to 5 errors. The results are also extended to column‐weight 4 LDPC codes of girth 6. Finally, we investigate eliminating of trapping sets of a column‐weight 3 LDPC code of girth 8 using the Gallager B decoding algorithm.

On the Construction of Low Error Floor LDPC Codes on Rectangular Lattices

In previous literature, arithmetically constrained slope sequences have been adopted for the construction of girth eight (3, k) LDPC codes on rectangular lattices. This method, however, creates a certain number of small trapping sets, which degrade overall performances. In this letter, an improved construction of a class of (3, k) LDPC codes on rectangular lattices is presented. Relevant constraints are derived to eliminate small trapping sets to improve performance in the error floor region. The check matrices of the proposed codes are introduced into quasi-cyclic structures, where experimental investigation shows favorable error rate performance over AWGN channels.

On the Construction of Structured LDPC Codes Free of Small Trapping Sets

We present a method to construct low-density parity-check (LDPC) codes with low error floors on the binary symmetric channel. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the trapping set ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum-product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four cycles.

Parity-Check Matrix Extension to Lower the Error Floors of Irregular LDPC Codes

Trapping sets have been identified as one of the main factors causing error floors of low-density parity-check (LDPC) codes at high SNR values. By adding several new rows to the original parity-check matrix, a novel method is proposed to eliminate small trapping sets in the LDPC code's Tanner graph. Based on this parity-check matrix extension, we design new codes with low error floors from the original irregular LDPC codes. Simulation results show that the proposed method can lower the error floors of irregular LDPC codes significantly at high SNR values over AWGN channels.

Constructing Short-Length Irregular LDPC Codes with Low Error Floor

Trapping sets (TSs) are known to cause error floors in regular and irregular low-density parity-check (LDPC) codes. By avoiding major error-contributing TSs during the code construction process, codes with low error floors can effectively be built. In it has been shown that TSs labeled as [w;u] are considered as being equivalent under the automorphism of the graph and are therefore contributing equally to the error floor. However, TSs with the same label [w;u] are not identical in general, particularly for the case of irregular LDPC codes. In this paper, we introduce a parameter e that can identify the number of "distinguishable" cycles in the connected subgraph induced by an elementary trapping set. Further, we propose a code construction algorithm, namely the Progressive-Edge-Growth Approximate-minimum-Cycle-Set-Extrinsic-message-degree (PEG-ACSE) method, that aims to avoid small elementary trapping sets (ETSs), particularly detrimental ETSs. We also develop theorems evaluating the minimum possible ETSs formed by PEG construction algorithms in general. We compare the characteristics of the codes built using the proposed method and those built using PEG-only or PEG-Approximate-minimum-Cycle-Extrinsic-message-degree (PEG-ACE) methods. Results from simulations show that the codes constructed using the proposed PEG-ACSE method produce lower error rates, particularly at the high signal-to-noise (SNR) region, compared with codes constructed using other PEG-based algorithms.

The Trapping Redundancy of Linear Block Codes

We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in any parity-check matrix of a given code and the size of its smallest trapping set. Trapping sets with certain parameter sizes are known to cause error-floors in the performance curves of iterative belief propagation (BP) decoders, and it is therefore important to identify decoding matrices that avoid such sets. Bounds on the trapping redundancy are obtained using probabilistic and constructive methods, and the analysis covers both general and elementary trapping sets. Numerical values for these bounds are computed for the [2640, 1320] Margulis code and the class of projective geometry codes, and compared with some new code-specific trapping set size estimates.