Generalized Low-density Parity-check Research Articles

Improving the Thresholds of Generalized LDPC Codes With Convolutional Code Constraints

CC-GLPDC codes are a class of generalized low-density parity-check (GLDPC) codes where the constraint nodes (CNs) represent convolutional codes. This allows for efficient decoding in the trellis with the forward-backward algorithm, and the strength of the component codes easily can be controlled by the encoder memory without changing the graph structure. In this letter, we extend the class of CC-GLDPC codes by introducing different types of irregularity at the CNs and investigating their effect on the BP and MAP decoding thresholds for the binary erasure channel (BEC). For the considered class of codes, an exhaustive grid search is performed to find the BP-optimized and MAP-optimized ensembles and compare their thresholds with the regular ensemble of the same design rate. The results show that irregularity can significantly improve the BP thresholds, whereas the thresholds of the MAP-optimized ensembles are only slightly different from the regular ensembles. Simulation results for the AWGN channel are presented as well and compared to the corresponding thresholds.

Trapping and Absorbing Set Enumerators for Irregular Generalized Low-Density Parity-Check Code Ensembles

The definitions of (elementary) trapping and (fully) absorbing sets are extended to generalized low-density parity-check (GLDPC) codes. In particular, the fully absorbing sets are stable under the bit flipping algorithm. The finite-length (elementary) trapping and (fully) absorbing set enumerators for unstructured irregular GLDPC code ensembles are derived using generating functions. An efficient method to evaluate the asymptotic trapping and (fully) absorbing set distributions is presented, which requires solving a system of equations. Using this method, the asymptotic distribution of trapping and (fully) absorbing sets are evaluated for some example GLDPC code ensembles. Experimental results show that the proposed definitions yield graph structures that are harmful for bit flipping decoders. The generating function approach is also used to derive the finite-length and asymptotic trapping and (elementary) absorbing set enumerators for non-binary irregular LDPC code ensembles.

The Cycle-Concentrating PEG Algorithm for Protograph Generalized LDPC Codes

The Cycle-Concentrating PEG Algorithm for Protograph Generalized LDPC Codes

Neural Min-Sum Decoding for Generalized LDPC Codes

In this letter, we investigate the min-sum (MS) and neural MS (NMS) decoding algorithms for generalized low-density parity-check (GLDPC) codes. Although the MS decoder is much simpler than the a posteriori probability (APP) decoder commonly used for GLDPC codes, the MS decoder has not been considered mainly due to its inferior decoding performance. However, we show that the performance can be improved by i) employing the NMS decoding algorithm and ii) optimizing the component parity check matrix (PCM). For the four representative short GLDPC codes in the literature, experimental results show that the NMS decoding performance with the optimized component PCM significantly outperforms the MS decoding performance and even outperforms the APP decoding performance for some cases.

Constant-Overhead Quantum Error Correction with Thin Planar Connectivity.

Quantum low density parity check (LDPC) codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, naïve layouts couple many distant qubits with crossing connections which could be hard to build in hardware and could result in performance-degrading crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their Tanner graphs into a small number of planar layers. Each layer contains long-range connections which do not cross. For any Calderbank-Shor-Steane code with a degree-δ Tanner graph, we design stabilizer measurement circuits with depth at most (2δ+2) using at most ⌈δ/2⌉ layers. We observe a circuit-noise threshold of 0.28% for a positive-rate code family using 49 physical qubits per logical qubit. For a physical error rate of 10^{-4}, this family reaches a logical error rate of 10^{-15} using fourteen times fewer physical qubits than the surface code.

Construction of Protograph-Based Partially Doped Generalized LDPC Codes

In this paper, we propose a new code design technique, called partial doping, for protograph-based generalized low-density parity-check (GLDPC) codes. While the conventional construction method of protograph-based GLDPC codes is to replace some single parity-check (SPC) nodes with generalized constraint (GC) nodes applying to multiple variable nodes (VNs) that are connected in the protograph, the proposed technique can select any VNs in the protograph to be protected by GC nodes. In other words, the partial doping technique facilitates finer tuning of doping, which in turn enables a sophisticated code optimization with higher degree of freedom. We construct the proposed partially doped GLDPC (PD-GLDPC) codes using the partial doping technique and optimize the PD-GLDPC codes by the protograph extrinsic information transfer (PEXIT) analysis. In addition, we propose a condition guaranteeing the linear minimum distance growth of the PD-GLDPC codes and use the condition for the optimization. Experimental results show that the optimized PD-GLDPC codes outperform the conventional GLDPC codes and have competitive performance compared to the state-of-the-art protograph-based LDPC codes without the error floor phenomenon over the binary erasure channel (BEC).

Refined Reliability Combining for Binary Message Passing Decoding of Product Codes

We propose a novel soft-aided iterative decoding algorithm for product codes (PCs). The proposed algorithm, named iterative bounded distance decoding with combined reliability (iBDD-CR), enhances the conventional iterative bounded distance decoding (iBDD) of PCs by exploiting some level of soft information. In particular, iBDD-CR can be seen as a modification of iBDD where the hard decisions of the row and column decoders are made based on a reliability estimate of the BDD outputs. The reliability estimates are derived using extrinsic message passing for generalized low-density-parity check (GLDPC) ensembles, which encompass PCs. We perform a density evolution analysis of iBDD-CR for transmission over the additive white Gaussian noise channel for the GLDPC ensemble. We consider both binary transmission and bit-interleaved coded modulation with quadrature amplitude modulation.We show that iBDD-CR achieves performance gains up to $0.51$ dB compared to iBDD with the same internal decoder data flow. This makes the algorithm an attractive solution for very high-throughput applications such as fiber-optic communications.

Spatially Coupled Generalized LDPC Codes: Asymptotic Analysis and Finite Length Scaling

Generalized low-density parity-check (GLDPC) codes are a class of LDPC codes in which the standard single parity check (SPC) constraints are replaced by constraints defined by a linear block code. These stronger constraints typically result in improved error floor performance, due to better minimum distance and trapping set properties, at a cost of some increased decoding complexity. In this paper, we study spatially coupled generalized low-density parity-check (SC-GLDPC) codes and present a comprehensive analysis of these codes, including: (1) an iterative decoding threshold analysis of SC-GLDPC code ensembles demonstrating capacity approaching thresholds via the threshold saturation effect; (2) an asymptotic analysis of the minimum distance and free distance properties of SC-GLDPC code ensembles, demonstrating that the ensembles are asymptotically good; and (3) an analysis of the finite-length scaling behavior of both GLDPC block codes and SC-GLDPC codes based on a peeling decoder (PD) operating on a binary erasure channel (BEC). Results are compared to GLDPC block codes, and the advantages and disadvantages of SC-GLDPC codes are discussed.

Extremal absorbing sets in low-density parity-check codes

Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of $ b $ for a given $ a $ for which an $ (a,b) $-absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.

Low Complexity Generalized-LDPC Decoder Based on Trellis Splicing

The high reliability short length low-density parity-check (LDPC) codes for Ultra Reliable Low Latency Communication (URLLC) are particularly important to future wireless communication systems. Even though the error floor region performance can be improved by using generalized-LDPC (G-LDPC) decoder, its penalty of increasing complexity can not be ignored. In this letter, we invoke the early stop criterion and the trellis splicing (TS) method to G-LDPC decoder. Hence, a G-LDPC-TS decoder is proposed. The proposed method can effectively reduce the number of variable nodes (VNs) involved in BCJR (Bahl-Cocke-Jelinek-Raviv) decoding of G-LDPC; thus, it yields a lower computational complexity. The results confirm that our G-LDPC-TS decoder can achieve a significant complexity reduction without sacrificing the performance.

Binary Message Passing Decoding of Product-Like Codes

We propose a novel binary message passing decoding algorithm for product-like codes based on bounded distance decoding (BDD) of the component codes. The algorithm, dubbed iterative BDD with scaled reliability (iBDD-SR), exploits the channel reliabilities and is therefore soft in nature. However, the messages exchanged by the component decoders are binary (hard) messages, which significantly reduces the decoder data flow. The exchanged binary messages are obtained by combining the channel reliability with the BDD decoder output reliabilities, properly conveyed by a scaling factor applied to the BDD decisions. We perform a density evolution analysis for generalized low-density parity-check (GLDPC) code ensembles and spatially coupled GLDPC code ensembles, from which the scaling factors of the iBDD-SR for product and staircase codes, respectively, can be obtained. For the white additive Gaussian noise channel, we show performance gains up to 0.29 dB and 0.31 dB for product and staircase codes compared to conventional iterative BDD (iBDD) with the same decoder data flow. Furthermore, we show that iBDD-SR approaches the performance of ideal iBDD that prevents miscorrections.

A Probabilistic Peeling Decoder to Efficiently Analyze Generalized LDPC Codes Over the BEC

In this paper, we analyze the tradeoff between coding rate and asymptotic performance of a class of generalized low-density parity-check (GLDPC) codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. The rate of the GLDPC ensemble is bounded using classical results on linear block codes, namely Hamming bound and Varshamov bound. We also study the impact of the decoding method used at GC nodes. To incorporate both bounded-distance (BD) and Maximum Likelihood (ML) decoding at GC nodes into our analysis without resorting on multi-edge type of degree distributions (DDs), we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code as well as its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as density evolution (DE) or the differential equation method. Furthermore, for a class of GLDPC ensembles, we demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. In all cases, we show that a large fraction of GC nodes is required to reduce the original gap to capacity, but the optimal fraction is strictly smaller than one. We then consider techniques to further reduce the gap to capacity by means of random puncturing, and the inclusion of a certain fraction of generalized variable nodes in the graph.

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

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.

Spatially Coupled Generalized Low-Density Parity-Check Codes Over Class-A Impulsive Noise Channels

In this paper, a class of spatially coupled generalized low-density parity-check (SC-GLDPC) codes with minimum coupling width is introduced to mitigate the Class-A impulsive noise. Codes with minimum coupling width are beneficial to simple code structure, short window decoding delay, and small rate-loss. The SC-GLDPC code is constructed by coupling the multiple identical generalized low-density parity-check (GLDPC) codes. Decoding thresholds over impulsive channels are derived by a recursive extrinsic information transfer function approach and are further estimated by the fast prediction approach. An edge spreading optimization method is proposed to design SC-GLDPC codes with minimum coupling width. The simulation results show that the optimized SC-GLDPC codes with minimum coupling width can show capacity-approaching property over impulsive channels. Benefitting from spatial coupling and generalized check nodes the SC-GLDPC codes show better bit error rate performance over impulsive channels, compared with GLDPC codes and spatially coupled low-density parity-check (SC-LDPC) codes.

Generalized LDPC Codes for Ultra Reliable Low Latency Communication in 5G and Beyond

Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. In this paper, a practical construction of quasi-cyclic GLDPC codes is proposed, where the proportion of generalized constraints is determined by an asymptotic analysis. We analyze the complexity and performance of the message passing decoder with various update rules (including standard full-precision sum-product and min-sum algorithms) and quantization schemes for a GLDPC code over the additive white Gaussian noise (AWGN) channel and present a constraint-to-variable update rule based on the specific codewords of the component codes. The block error rate performance of the GLDPC codes, combined with a complementary outer code, is shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication regime over an AWGN channel with quadrature PSK modulation.

GLDPC-Staircase AL-FEC codes: a fundamental study and new results

This paper provides fundamentals in the design and analysis of Generalized Low-Density Parity Check (GLDPC)-Staircase codes over the erasure channel. These codes are constructed by extending an LDPC-Staircase code (base code) using Reed-Solomon (RS) codes (outer codes) in order to benefit from more powerful decoders. The GLDPC-Staircase coding scheme adds, in addition to the LDPC-Staircase repair symbols, extra-repair symbols that can be produced on demand and in large quantities, which provides small rate capabilities. Therefore, these codes are extremely flexible as they can be tuned to behave either like predefined rate LDPC-Staircase codes at one extreme, or like a single RS code at another extreme, or like small rate codes. Concerning the code design, we show that RS codes with “quasi” Hankel matrix-based construction fulfill the desired structure properties, and that a hybrid (IT/RS/ML) decoding is feasible that achieves maximum likelihood (ML) correction capabilities at a lower complexity. Concerning performance analysis, we detail an asymptotic analysis method based on density evolution (DE), extrinsic information transfer (EXIT), and the area theorem. Based on several asymptotic and finite length results, after selecting the optimal internal parameters, we demonstrate that GLDPC-Staircase codes feature excellent erasure recovery capabilities, close to that of ideal codes, both with large and very small objects. From this point of view, they outperform LDPC-Staircase and Raptor codes and achieve correction capabilities close to those of RaptorQ codes. Therefore, all these results make GLDPC-Staircase codes a universal Application-Layer FEC (AL-FEC) solution for many situations that require erasure protection such as media streaming or file multicast transmission.

A General Non-Binary LDPC Code Optimization Framework Suitable for Dense Flash Memory and Magnetic Storage

Transmission channels underlying modern dense storage systems, e.g., Flash memory and magnetic recording (MR) systems, significantly differ from canonical channels, like additive white Gaussian noise (AWGN) channels. While existing low-density parity-check (LDPC) codes optimized for symmetric, AWGN-like channels are being actively considered for Flash applications, we demonstrate that, due to channel asymmetry, such approaches are inadequate. We introduce a refined definition of absorbing sets, which we call general absorbing sets of type two (GASTs) , and study the combinatorial properties of GASTs. We then present the weight consistency matrix (WCM) , which succinctly captures key properties in a GAST. Furthermore, we show how to customize the WCM definition such that it suits other special subclasses of GASTs. Based on these new concepts, we then develop a new, general combinatorial code optimization framework, which we call the WCM framework , and demonstrate its effectiveness on the realistic highly-asymmetric normal-Laplace mixture (NLM) Flash channel. Moreover, we show that our framework can be customized to optimize non-binary LDPC (NB-LDPC) codes for other asymmetric channels, channels with memory (incorporated in MR systems), and canonical symmetric channels. For all the channels we have simulated NB-LDPC codes over, the codes optimized using the WCM framework enjoy at least 1 order, and up to nearly 2 orders of magnitude performance gain in the uncorrectable bit error rate (UBER) or the frame error rate (FER) relative to the unoptimized codes. Our simulations also show that codes optimized for symmetric channels are not the best choice for asymmetric channels.

Construction of Type‐II QC LDPC Codes Based on Perfect Cyclic Difference Set

Quasi-cyclic (QC) Low-density paritycheck (LDPC) codes are constructed from combination of weight-0 (null matrix) and Weight-2 (W2) Circulant matrix (CM), which can be seen as a special case of the general type-II QC LDPC codes. The shift matrix of the codes is built on the basis of one integer sequence, called perfect Cyclic difference set (CDS), which guarantees the girth of the code at least six. Simulation results show that the codes can perform well in comparison with a variety of other LDPC codes. They have excellent error floor and decoding convergence characteristics.

이진 순환 부호를 쓰는 GLDPC 부호의 수평-수직 결합 직렬 복호

It is well known that serial belief propagation (BP) decoding for low-density parity-check (LDPC) codes achieves faster convergence without any increase of decoding complexity per iteration and bit error rate (BER) performance loss than standard parallel BP (PBP) decoding. Serial BP (SBP) decoding, such as horizontal SBP (H-SBP) decoding or vertical SBP (V-SBP) decoding, updates check nodes or variable nodes faster than standard PBP decoding within a single iteration. In this paper, we propose combined horizontal-vertical SBP (CHV-SBP) decoding. By the same reasoning, CHV-SBP decoding updates check nodes or variable nodes faster than SBP decoding within a serialized step in an iteration. CHV-SBP decoding achieves faster convergence than H-SBP or V-SBP decoding. We compare these decoding schemes in details. We also show in simulations that the convergence rate, in iterations, for CHV-SBP decoding is about <TEX>$\frac{1}{6}$</TEX> of that for standard PBP decoding, while the convergence rate for SBP decoding is about <TEX>$\frac{1}{2}$</TEX> of that for standard PBP decoding. In simulations, we use recently proposed generalized LDPC (GLDPC) codes with binary cyclic codes (BCC).

Multiple component codes based generalized LDPC codes for high-speed optical transport.

A class of generalized low-density parity-check (GLDPC) codes suitable for optical communications is proposed, which consists of multiple local codes. It is shown that Hamming, BCH, and Reed-Muller codes can be used as local codes, and that the maximum a posteriori probability (MAP) decoding of these local codes by Ashikhmin-Lytsin algorithm is feasible in terms of complexity and performance. We demonstrate that record coding gains can be obtained from properly designed GLDPC codes, derived from multiple component codes. We then show that several recently proposed classes of LDPC codes such as convolutional and spatially-coupled codes can be described using the concept of GLDPC coding, which indicates that the GLDPC coding can be used as a unified platform for advanced FEC enabling ultra-high speed optical transport. The proposed class of GLDPC codes is also suitable for code-rate adaption, to adjust the error correction strength depending on the optical channel conditions.