In this paper, we propose a non-uniform windowed decoder for multi-dimensional spatially-coupled LDPC (MD-SC-LDPC) codes over the binary erasure channel. An MD-SC-LDPC code is constructed by connecting together several SC-LDPC codes into one larger code that provides major benefits over a variety of channel models. In general, SC codes allow for low-latency windowed decoding. While a standard windowed decoder can be naively applied, such an approach does not fully utilize the unique structure of MD-SC-LDPC codes. In this paper, we propose and analyze a novel non-uniform decoder to provide more flexibility between latency and reliability. Our theoretical derivations and empirical results show that our non-uniform decoder greatly improves upon the standard windowed decoder in terms of design flexibility, latency, and complexity.
In this letter, we propose a reliability-based windowed decoding scheme for spatially-coupled (SC) low-density parity-check (LDPC) codes. To mitigate the error propagation along the sliding windowed decoder of the SC LDPC codes, a partial message reservation (PMR) method is proposed where only the reliable messages generated in the previous decoding window are reserved for the next decoding window. We also propose a partial syndrome check (PSC) stopping rule for each decoding window, in which only the complete VNs are checked. Simulation results show that our proposed scheme significantly impr…
We consider a windowed decoding scheme for LDPC convolutional codes that is based on the belief-propagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP dec…
In this work, we analyze efficient window shift schemes for windowed decoding of spatially coupled low-density parity-check (SC-LDPC) codes, which is known to yield close-tooptimal decoding results when compared to full belief propagation (BP) decoding. However, a drawback of windowed decoding is that either a significant amount of window updates are required leading to unnecessary high decoding complexity or the decoder suffers from sporadic burst-like error patterns, causing a decoder stall. To tackle this effect and, thus, to reduce the average decoding complexity, the basic idea is to enab…
In this paper we study the iterative decoding threshold performance of non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code ensembles for both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC), with particular emphasis on windowed decoding (WD). We consider both (2,4)-regular and (3,6)-regular NB-SC-LDPC code ensembles constructed using protographs and compute their thresholds using protograph versions of NB density evolution and NB extrinsic information transfer analysis. For these code ensembles, we show that WD of NB-SC-…
In this paper we consider the generalization of binary spatially coupled low-density parity-check (SC-LDPC) codes to finite fields GF$(q)$, $q\geq 2$, and develop design rules for $q$-ary SC-LDPC code ensembles based on their iterative belief propagation (BP) decoding thresholds, with particular emphasis on low-latency windowed decoding (WD). We consider transmission over both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC) and present results for a variety of $(J,K)$-regular SC-LDPC code ensembles constructed over GF$(q)$ using protographs…
Spatially Coupled Low-Density Parity-Check (SC-LDPC) codes offer excellent decoding performance and can be elegantly decoded with a Windowed Decoder (WD). We determine an efficient WD configuration with low control overhead. For fair comparisons, we normalize all configurations to the same maximal computational complexity, which is an important measure of the decoding effort in packet-based data communication systems. We determine an optimized configuration from a joint evaluation of the window size, the window update strategy, and parity check-based Early Termination (ET). Firstly, we use a v…
We consider spatially coupled low-density parity-check codes with finite smoothing parameters. A finite smoothing parameter is important for designing practical codes that are decoded using low-complexity windowed decoders. By optimizing the amount of coupling between spatial positions, we show that we can construct codes with excellent thresholds and small rate loss, even with the lowest possible smoothing parameter and large variable node degrees, which are required for low error floors. We also establish that the decoding convergence speed is faster with non-uniformly coupled codes, which w…
In this paper, we investigate the joint design of channel and network coding in bi-directional relaying systems and propose a combined low complexity physical network coding and LDPC decoding scheme. For the same LDPC codes employed at both source nodes, we show that the relay can decodes the network coded codewords from the superimposed signal received from the BPSK-modulated multiple-access channel. Simulation results shown that this novel joint physical network coding and LDPC decoding method outperforms the existing MMSE network coding and LDPC decoding method over AWGN and complex MAC cha…
This paper is concerned with linear superposition systems in which all components of the superimposed signal are coded with an identical binary low-density parity-check (LDPC) code.
Bliss schemes of a run length limited (RLL) codec in combination with an LDPC codec, generate LDPC parity bits over a systematic sequence of RLL channel bits that are inherently redundant as they satisfy e.g. a $d=1$ minimum run length constraint. That is the subsequences consisting of runs of length $d=1$, viz. $...010...$ and $...101...$, cannot occur. We propose to use this redundancy during LDPC decoding in a Bliss scheme by introducing additional $d$-constraint nodes in the factor graph used by the LDPC decoder. The messages sent from these new nodes to the variable or codeword bit nodes …
Delayed bit-interleaved coded modulation (DBICM) generalizes bit-interleaved coded modulation (BICM) by modulating differently delayed sub-blocks of codewords onto the same signals. DBICM improves transmission reliability over BICM due to its capability of detecting undelayed sub-blocks with the extrinsic information of the decoded delayed sub-blocks. In this work, we propose a novel windowed decoding algorithm for DBICM, which uses the extrinsic information of both the decoded delayed and undelayed sub-blocks, to improve the detection on all sub-blocks. Numerical results show that the propose…
In this paper, we consider the performance of the Noisy Gradient Descent Bit Flipping (NGDBF) algorithm under re-decoding of failed frames. NGDBF is a recent algorithm that uses a non-deterministic gradient descent search to decode low-density parity check (LDPC) codes. The proposed re-decode procedure obtains improved performance because the perturbations are independent at each re-decoding phase, therefore increasing the likelihood of successful decoding. We examine the benefits of re-decoding for an LDPC code from the IEEE 802.3an standard, and find that only a small fraction of re-decoded …
A new method for low-complexity near-maximum-likelihood (ML) decoding of low-density parity-check (LDPC) codes over the additive white Gaussian noise channel is presented. The proposed method termed belief-propagation--list erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits performed in case of BP decoding failure. A strategy of introducing erasures into the received vector and a new erasure decoding algorithm are proposed. The new erasure decoding algorithm, called list erasure decoding, combines ML decoding over the BEC with list decoding applied if the ML decoder…
In this paper, we present a low-complexity joint detection-decoding algorithm for nonbinary LDPC codedmodulation systems. The algorithm combines hard-decision decoding using the message-passing strategy with the signal detector in an iterative manner. It requires low computational complexity, offers good system performance and has a fast rate of decoding convergence. Compared to the q-ary sum-product algorithm (QSPA), it provides an attractive candidate for practical applications of q-ary LDPC codes.
Layered decoding is well appreciated in Low-Density Parity-Check (LDPC) decoder implementation since it can achieve effectively high decoding throughput with low computation complexity. This work, for the first time, addresses low complexity column-layered decoding schemes and VLSI architectures for multi-Gb/s applications. At first, the Min-Sum algorithm is incorporated into the column-layered decoding. Then algorithmic transformations and judicious approximations are explored to minimize the overall computation complexity. Compared to the original column-layered decoding, the new approach ca…
We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the propose…
We introduce a novel algorithm for decoding binary linear codes by linear programming. We build on the LP decoding algorithm of Feldman et al. and introduce a post-processing step that solves a second linear program that reweights the objective function based on the outcome of the original LP decoder output. Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. We also show significant empirical performance gains for the reweighted LP decoding algorithm with very small additional computational complexity.
A novel class of bit-flipping (BF) algorithms for decoding low-density parity-check (LDPC) codes is presented. The proposed algorithms, which are called gradient descent bit flipping (GDBF) algorithms, can be regarded as simplified gradient descent algorithms. Based on gradient descent formulation, the proposed algorithms are naturally derived from a simple non-linear objective function.
In this paper a new message passing algorithm, which takes advantage of both tree-based re-parameterization and the knowledge of short cycles, is introduced for the purpose of decoding LDPC codes with short block lengths. The proposed algorithm is called variable factor appearance probability belief propagation (VFAP-BP) algorithm and is suitable for wireless communications applications, where both good decoding performance and low-latency are expected. Our simulation results show that the VFAP-BP algorithm outperforms the standard BP algorithm and requires a significantly smaller number of it…
We consider the decoding of LDPC codes over GF(q) with the low-complexity majority algorithm from [1]. A modification of this algorithm with multiple thresholds is suggested. A lower estimate on the decoding radius realized by the new algorithm is derived. The estimate is shown to be better than the estimate for a single threshold majority decoder. At the same time the transition to multiple thresholds does not affect the order of complexity.
In this paper, we focus on decoding nonbinary low-density parity-check (LDPC) codes in Galois fields of characteristic two via the proximal alternating direction method of multipliers (proximal-ADMM). By exploiting Flanagan/Constant-Weighting embedding techniques and the decomposition technique based on three-variables parity-check equations, two efficient proximal-ADMM decoders for nonbinary LDPC codes are proposed. We show that both of them are theoretically guaranteed convergent to some stationary point of the decoding model and either of their computational complexities in each proximal-AD…
Owing to their capacity-achieving performance and low encoding and decoding complexity, polar codes have drawn much research interests recently. Successive cancellation decoding (SCD) and belief propagation decoding (BPD) are two common approaches for decoding polar codes. SCD is sequential in nature while BPD can run in parallel. Thus BPD is more attractive for low latency applications. However BPD has some performance degradation at higher SNR when compared with SCD. Concatenating LDPC with Polar codes is one popular approach to enhance the performance of BPD , where a short LDPC code is use…
An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly \emph{punctured} output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded \emph{complexity}. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The st…
The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density parity-check (LDPC) codes from two random ensembles is studied. The theoretical average spectrum of the Gallager ensemble is computed by using a recurrent procedure and compared to the empirically found average spectrum for the same ensemble as well as to the empirical average spectrum of the Richardson-Urbanke ensemble and spectra of selected codes from both ensembles. Distance properties of the random codes from the Gallager ensemble are discussed. A tightened union-type upper boun…
In this paper, we show that Quasi-Cyclic LDPC codes can efficiently accommodate the hybrid iterative/ML decoding over the binary erasure channel. We demonstrate that the quasi-cyclic structure of the parity-check matrix can be advantageously used in order to significantly reduce the complexity of the ML decoding. This is achieved by a simple row/column permutation that transforms a QC matrix into a pseudo-band form. Based on this approach, we propose a class of QC-LDPC codes with almost ideal error correction performance under the ML decoding, while the required number of row/symbol operations…
The power consumption of the integrated circuit is becoming a significant burden, particularly for large-scale signal processing tasks requiring high throughput. The decoding process of LDPC codes is such a heavy signal processing task that demands power efficiency and higher decoding throughput. A promising approach to reducing both power and latency of a decoding process is to use an analog circuit instead of a digital circuit. This paper investigates a continuous-time gradient flow-based approach for decoding LDPC codes, which employs a potential energy function similar to the objective fun…
Cooperative optimization is a new way for finding global optima of complicated functions of many variables. It has some important properties not possessed by any conventional optimization methods. It has been successfully applied in solving many large scale optimization problems in image processing, computer vision, and computational chemistry. This paper shows the application of this optimization principle in decoding LDPC codes, which is another hard combinatorial optimization problem. In our experiments, it significantly out-performed the sum-product algorithm, the best known method for dec…
In this paper, we provide necessary and sufficient conditions for a column-weight-three LDPC code to correct three errors when decoded using Gallager A algorithm. We then provide a construction technique which results in a code satisfying the above conditions. We also provide numerical assessment of code performance via simulation results.
In this paper we present a thorough analysis of non binary LDPC codes over the binary erasure channel. First, the decoding of non binary LDPC codes is investigated. The proposed algorithm performs on-the-fly decoding, i.e. it starts decoding as soon as the first symbols are received, which generalizes the erasure decoding of binary LDPC codes. Next, we evaluate the asymptotical performance of ensembles of non binary LDPC codes, by using the density evolution method. Density evolution equations are derived by taking into consideration both the irregularity of the bipartite graph and the probabi…
We propose a novel optimization-based decoding algorithm for LDPC-coded massive MIMO channels. The proposed decoding algorithm is based on a proximal gradient method for solving an approximate maximum a posteriori (MAP) decoding problem. The key idea is the use of a code-constraint polynomial penalizing a vector far from a codeword as a regularizer in the approximate MAP objective function. The code proximal operator is naturally derived from code-constraint polynomials. The proposed algorithm, called proximal decoding, can be described by a simple recursion consisting of the gradient descent …
The Alternating Direction Method of Multipliers (ADMM) decoding of Low Density Parity Check (LDPC) codes has received many attentions due to its excellent performance at the error floor region. In this paper, we develop a parameter-free decoder based on Linear Program (LP) decoding by replacing the binary constraint with the intersection of a box and an $\ell_p$ sphere. An efficient $\ell_2$-box ADMM is designed to handle this model in a distributed fashion. Numerical experiments demonstrate that our decoder attains better adaptability to different Signal-to-Noise Ratio and channels.
We address the problem of the joint sequence detection in partial-response (PR) channels and decoding of low-density parity-check (LDPC) codes. We model the PR channel and the LDPC code as a combined inference problem. We present for the first time the derivation of the belief propagation (BP) equations that allow the simultaneous detection and decoding of a LDPC codeword in a PR channel. To accomplish this we follow an approach from statistical mechanics, in which the Bethe free energy is minimized with respect to the beliefs on the nodes of the PR-LDPC graph. The equations obtained are expli…
In his thesis, Wiberg showed the existence of thresholds for families of regular low-density parity-check codes under min-sum algorithm decoding. He also derived analytic bounds on these thresholds. In this paper, we formulate similar results for linear programming decoding of regular low-density parity-check codes.
Iterative decoding of non-binary LDPC codes is currently performed using either the Sum-Product or the Min-Sum algorithms or slightly different versions of them. In this paper, several low-complexity quasi-optimal iterative algorithms are proposed for decoding non-binary codes. The Min-Max algorithm is one of them and it has the benefit of two possible LLR domain implementations: a standard implementation, whose complexity scales as the square of the Galois field's cardinality and a reduced complexity implementation called selective implementation, which makes the Min-Max decoding very attract…
Spatially coupled low-density parity-check (SC-LDPC) codes are sparse graph codes that have recently become of interest due to their capacity-approaching performance on memoryless binary input channels. In this paper, we unify all existing SC-LDPC code construction methods under a new generalized description of SC-LDPC codes based on algebraic lifts of graphs. We present an improved low-complexity counting method for the special case of $(3,3)$-absorbing sets for array-based SC-LDPC codes, which we then use to optimize permutation assignments in SC-LDPC code construction. We show that codes co…
Sparse code multiple access (SCMA) is a promising multiplexing approach to achieve high system capacity. In this paper, we develop a novel iterative detection and decoding scheme for SCMA systems combined with Low-density Parity-check (LDPC) decoding. In particular, we decompose the output of the message passing algorithm (MPA) based SCMA multiuser detection into intrinsic part and prior part. Then we design a joint detection and decoding scheme which iteratively exchanges the intrinsic information between the detector and the decoder, yielding a satisfied performance gain. Moreover, the propo…
Protograph-based, off-the-shelf low-density parity-check (LDPC) codes are optimized for higher-order modulation and quantized sum-product decoders. As an example, for the recently proposed LDPC code from the upcoming IEEE 802.3ca standard for passive optical networks (PONs), an optimized mapping of the bit channels originating from bit-metric decoding to the protograph variable nodes gains 0.4 dB and 0.3 dB at a bit-error rate of 1e-6 for shaped and uniform signaling, respectively. Furthermore, the clipping value for a quantized sum-product LDPC decoder is optimized via discretized density evo…
We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes. Belief propagation (BP) decoding on the conventional factor graph is equivariant to the quasi-cyclic automorphisms and therefore prevents gains by AED. However, by applying small modifications to the parity-check matrix at the receiver side, we can break the symmetry without changing the code at the transmitter. This way, we can leverage a gain in error-correcting performance using an ensemble of identical BP decoders, without increasing the worst-case decoding latency. The proposed met…
Min-Sum decoding is widely used for decoding LDPC codes in many modern digital video broadcasting decoding due to its relative low complexity and robustness against quantization error. However, the suboptimal performance of the Min-Sum affects the integrated performance of wireless receivers. In this paper, we present the idea of adapting the scaling factor of the Min-Sum decoder with iterations through a simple approximation. For the ease of implementation the scaling factor can be changed in a staircase fashion. The stair step is designed to optimize the decoder performance and the required …
This paper presents a theoretical study of a new type of LDPC codes motivated by practical storage applications. LDPCL codes (suffix L represents locality) are LDPC codes that can be decoded either as usual over the full code block, or locally when a smaller sub-block is accessed (to reduce latency). LDPCL codes are designed to maximize the error-correction performance vs. rate in the usual (global) mode, while at the same time providing a certain performance in the local mode. We develop a theoretical framework for the design of LDPCL codes. Our results include a design tool to construct an L…
The asymptotic iterative decoding performances of low-density parity-check (LDPC) codes using min-sum (MS) and sum-product (SP) decoding algorithms on memoryless binary-input output-symmetric (MBIOS) channels are analyzed in this paper. For MS decoding, the analysis is done by upper bounding the bit error probability of the root bit of a tree code by the sequence error probability of a subcode of the tree code assuming the transmission of the all-zero codeword. The result is a recursive upper bound on the bit error probability after each iteration. For SP decoding, we derive a recursively dete…
In practice, LDPC codes are decoded using message passing methods. These methods offer good performance but tend to converge slowly and sometimes fail to converge and to decode the desired codewords correctly. Recently, tree-reweighted message passing methods have been modified to improve the convergence speed at little or no additional complexity cost. This paper extends this line of work and proposes a new class of locally optimized reweighting strategies, which are suitable for both regular and irregular LDPC codes. The proposed decoding algorithm first splits the factor graph into subgraph…
Spatially coupled codes have been of interest recently owing to their superior performance over memoryless binary-input channels. The performance is good both asymptotically, since the belief propagation thresholds approach capacity, as well as for finite lengths, since degree-2 variables that result in high error floors can be completely avoided. However, to realize the promised good performance, one needs large blocklengths. This in turn implies a large latency and decoding complexity. For the memoryless binary erasure channel, we consider the decoding of spatially coupled codes through a wi…
In this paper, we propose to study and optimize a very general class of LDPC codes whose variable nodes belong to finite sets with different orders. We named this class of codes Hybrid LDPC codes. Although efficient optimization techniques exist for binary LDPC codes and more recently for non-binary LDPC codes, they both exhibit drawbacks due to different reasons. Our goal is to capitalize on the advantages of both families by building codes with binary (or small finite set order) and non-binary parts in their factor graph representation. The class of Hybrid LDPC codes is obviously larger than…
We present a novel optimization-based decoding algorithm for LDPC codes that is suitable for hardware architectures specialized to feed-forward neural networks. The algorithm is based on the projected gradient descent algorithm with a penalty function for solving a non-convex minimization problem. The proposed algorithm has several internal parameters such as step size parameters, a softness parameter, and the penalty coefficients. We use a standard tool set of deep learning, i.e., back propagation and stochastic gradient descent (SGD) type algorithms, to optimize these parameters. Several num…
We address noisy message-passing decoding of lowdensity parity-check (LDPC) codes over additive white Gaussian noise channels. Message-passing decoders in which certain processing units iteratively exchange messages are common for decoding LDPC codes. The exchanged messages are in general subject to internal noise in hardware implementation of these decoders. We model the internal decoder noise as additive white Gaussian noise (AWGN) degrading exchanged messages. Using Gaussian approximation of the exchanged messages, we perform a two-dimensional density evolution analysis for the noisy LDPC d…
Due to the speed limitation of the conventional bit-chosen strategy in the existing weighted bit flipping algorithms, a high-speed LDPC decoder cannot be realized. To solve this problem, we propose a fast weighted bit flipping (FWBF) algorithm. Specifically, based on the stochastic error bitmap of the received vector, a partially parallel bit-choose strategy is adopted to lower the delay of choosing the bit flipped. Because of its partially parallel structure, the novel strategy can be well incorporated into the LDPC decoder [1]. The analysis of the decoding delay demonstrates that, the decodi…
We study spatially coupled LDPC codes that allow access to sub-blocks much smaller than the full code block. Sub-block access is realized by a semi-global decoder that decodes a chosen target sub-block by only accessing the target, plus a prescribed number of helper sub-blocks adjacent in the code chain. This paper develops a theoretical methodology for analyzing the semi-global decoding performance of spatially coupled LDPC codes constructed from protographs. The main result shows that semi-global decoding thresholds can be derived from certain thresholds we define for the single-sub-block gr…
A new approach for combining non-binary low-density parity-check (NB-LDPC) codes with higher-order modulation and probabilistic amplitude shaping (PAS) is presented. Instead of symbol-metric decoding (SMD), a bit-metric decoder (BMD) is used so that matching the field order of the non-binary code to the constellation size is not needed, which increases the flexibility of the coding scheme. Information rates, density evolution thresholds and finite-length simulations show that the flexibility comes at no loss of performance if PAS is used.
A new type of spatially coupled low-density parity-check (SC-LDPC) codes motivated by practical storage applications is presented. SC-LDPCL codes (suffix 'L' stands for locality) can be decoded locally at the level of sub-blocks that are much smaller than the full code block, thus offering flexible access to the coded information alongside the strong reliability of the global full-block decoding. Toward that, we propose constructions of SC-LDPCL codes that allow controlling the trade-off between local and global correction performance. In addition to local and global decoding, the paper develo…
We study error bounds for linear programming decoding of regular LDPC codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly-exponential in the girth of the factor graph. For memoryless binary-input AWGN channel, we prove lower bounds on the threshold for regular LDPC codes whose factor graphs have logarithmic girth under LP-decoding. Specifically, we prove a lower bound of $σ=0.735$ (upper bound of $\frac{Eb}{N_0}=2.67$dB) on the threshold of $(3,6)$-regular LDPC codes whose factor graphs have logarithmic girth. Our …
Recent works showed how low-density parity-check (LDPC) erasure correcting codes, under maximum likelihood (ML) decoding, are capable of tightly approaching the performance of an ideal maximum-distance-separable code on the binary erasure channel. Such result is achievable down to low error rates, even for small and moderate block sizes, while keeping the decoding complexity low, thanks to a class of decoding algorithms which exploits the sparseness of the parity-check matrix to reduce the complexity of Gaussian elimination (GE). In this paper the main concepts underlying ML decoding of LDPC c…
Parameters of LDPC codes, such as minimum distance, stopping distance, stopping redundancy, girth of the Tanner graph, and their influence on the frame error rate performance of the BP, ML and near-ML decoding over a BEC and an AWGN channel are studied. Both random and structured LDPC codes are considered. In particular, the BP decoding is applied to the code parity-check matrices with an increasing number of redundant rows, and the convergence of the performance to that of the ML decoding is analyzed. A comparison of the simulated BP, ML, and near-ML performance with the improved theoretical …
