Tanner Graph Research Articles

LP decoding of codes with expansion parameter above 2/3

A code C⊆F2n is a (c,δ,ϵ)-expander code if it has a Tanner graph, where every variable node has degree c, and every subset of variable nodes L0 such that |L0|⩽δn has at least ϵc|L0| neighbors.Feldman et al. (2007) [3] proved that LP decoding corrects 3ϵ−22ϵ−1⋅(δn−1) errors of (c,δ,ϵ)-expander code, where ϵ>23+13c.In this paper, we provide a slight consolidation of their work and show that this result holds for every expansion parameter ϵ>23.

PN Code Acquisition Using Belief Propagation with Adaptive Parity Check Matrix

Pseudonoise (PN) code acquisition technique based on iterative message passing algorithm (iMPA) has been proposed due to its short acquisition time and low complexity. However, the cyclic and regular nature of constructed tanner graph makes it difficult to achieve promising performance. To address this problem, this correspondence proposes a new message passing algorithm based on adaptive parity check matrix. We find multiple sets of linear sparse constraints for PN sequence by squaring the generator polynomial. The topology of the graphic models as well as the parity check matrix is adapted every a few iterations to avoid local optima. The performance of proposed algorithm is evaluated in terms of detection probability. Simulation results show that this method provides more than 3 dB gains over iMPA with fixed parity check matrix.

Weighted-Bit-Flipping-Based Sequential Scheduling Decoding Algorithms for LDPC Codes

Low-density parity-check (LDPC) codes can be applied in a lot of different scenarios such as video broadcasting and satellite communications. LDPC codes are commonly decoded by an iterative algorithm called belief propagation (BP) over the corresponding Tanner graph. The original BP updates all the variable-nodes simultaneously, followed by all the check-nodes simultaneously as well. We propose a sequential scheduling algorithm based on weighted bit-flipping (WBF) algorithm for the sake of improving the convergence speed. Notoriously, WBF is a low-complexity and simple algorithm. We combine it with BP to obtain advantages of these two algorithms. Flipping function used in WBF is borrowed to determine the priority of scheduling. Simulation results show that it can provide a good tradeoff between FER performance and computation complexity for short-length LDPC codes.

Local-Optimality Guarantees Based on Paths for Optimal Decoding

This paper presents a unified analysis framework that captures recent advances in the study of local-optimality characterizations for codes on graphs. These local-optimality characterizations are based on combinatorial structures embedded in the Tanner graph of the code. Local optimality implies both unique maximum likelihood optimality and unique linear programming (LP) decoding optimality. Also, an iterative message-passing decoding algorithm is guaranteed to find the unique locally optimal codeword if one exists. We demonstrate an instance of this proof technique by considering a definition of local optimality that is based on the simplest combinatorial structures in Tanner graphs, namely, paths of length $h$. We apply the technique of local optimality to binary Tanner codes (including any low-density parity-check code, and in particular any irregular repeat-accumulate code with both even and odd repetition factors). Inverse polynomial bounds in the code length are proved on the word error probability of LP decoding for binary Tanner codes. When the local codes are restricted to single parity-check codes, these bounds also hold for decoding by a certain iterative message-passing decoding algorithm.

An LDPC coded cooperative MIMO scheme over Rayleigh fading channels with unknown channel state information

This paper describes a coded cooperative multiple-input multiple-output (MIMO) scheme, where structured low-density parity-check (LDPC) codes belonging to a family of repeat-accumulate (RA) codes are employed. The outage probability of the scheme over Rayleigh fading channels is deduced. In an unknown channel state information (CSI) scenario, adaptive transversal filters based on a spatio-temporal recursive least squares (ST-RLS) algorithm are adopted in the destination to realize receive diversity gain. Also, a joint ‘Min-Sum’ iterative decoding is effectively carried out in the destination. Such a decoding algorithm agrees with the bilayer Tanner graph that can be used to fully characterize two distinct structured LDPC codes employed by the source and relay. Simulation results verify the effectiveness of the adopted filter in the coded cooperative MIMO scheme. Theoretical analysis and numerical simulations show that the LDPC coded cooperative MIMO scheme can well combine cooperation diversity, multi-receive diversity, and channel coding gains, and clearly outperforms coded noncooperation schemes under the same conditions.

Joint iterative decoding for LDPC‐coded multi‐relay cooperation with receive multi‐antenna in the destination

This study proposes an low density party check (LDPC)-coded multi-relay cooperation with receive multi-antenna in the destination over a Rayleigh-fading channel, where maximal ratio combining and belief propagation algorithm (BP)-based joint iterative decoding based on the introduced multi-layer Tanner graph are effectively designed to detect and decode the corrupted received sequence at the destination. The overall code rate from the destination is derived mathematically in relation to all the constituent LDPC codes employed by source and multiple relays. Theoretical analysis and numerical simulation show that the proposed approach can well combine diversity and coding gains, which consequently result in significant advantage over the conventional cooperations under the same conditions.

量子高斯密钥分发的逆向数据协调

The party Alice in a cryptograph system transmits coherent quantum signals to the party Bob,obtaining a series of correlated continuous Gaussian variables.The process to distill a binary key out of the correlated Gaussian variables that they share is called continuous-variable quantum key distribution(CVQKD).Reconciliation in CVQKD,i.e.,to correct differences between the correlated variables,is the key issue discussed in the paper.Based on both sliced error correction(SEC) protocol and multilevel coding/multistage decoding(MLC/MSD),reconciliation is implemented by LDPC code-based Slepian-Wolf CODEC.According to the proposed multilevel Tanner graph which depicts message transition of information,multistage iterative intrinsic and extrinsic information belief-propagation(LLR-BP) update formulae are derived.Three decoding modes of MSD are compared experimentally in convergence and performance.Simulation results show that the proposed algorithm can reconcile reliably 10000 continuous quantum variables with efficiency of 92.3% when SNR of the channel is above 2.5 dB.

Performance Analysis of Iterative Decoding Algorithms with Memory over Memoryless Channels

Density evolution is often used to determine the performance of an ensemble of low-density parity-check (LDPC) codes under iterative message-passing algorithms. Conventional density evolution techniques over memoryless channels are based on the assumption that messages at iteration l are only a function of the messages at iteration l -1 and possibly the channel output. This assumption is valid for many algorithms such as standard belief propagation (BP) and min-sum (MS) algorithms. However, there are other important iterative algorithms such as successive relaxation (SR) versions of BP and MS, and differential decoding with binary message passing (DD-BMP) algorithm of Mobini et al., for which this assumption is not valid. The reason is the introduction of memory in these algorithms. In this work, we propose a model for iterative decoding algorithms with memory which covers SR and DD-BMP algorithms as special cases. Based on this model, we derive a Bayesian network for iterative algorithms with memory over memoryless channels and use this representation to analyze the performance of the algorithms using density evolution. The density evolution technique is developed based on truncating the memory of the decoding process and approximating it with a finite order Markov process, and can be implemented efficiently. As an example, we apply our technique to analyze the performance of DD-BMP on regular LDPC code ensembles, and make a number of interesting observations with regard to the performance/complexity tradeoff of DD-BMP in comparison with BP and MS algorithms. The model presented in this paper is based on certain simplifying assumptions about the memory structure of iterative algorithms such as the existence of memory only at the output of variable nodes in the code's Tanner graph rather than at both outputs of variable and check nodes. The Bayesian network framework introduced here however, can still be used to analyze the more general scenarios.

Low-Complexity Z_4 LDPC Code Design under a Gaussian Approximation

In this paper we propose low complexity LDPC code design and decoding in Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> which may be useful to combat phase ambiguities in wireless links affected by strong phase noise. We approximate messages exchanged on the Tanner graph using separable probability density functions. This allows a substantial reduction of decoder memory and complexity, with a negligible performance penalty, compared to ideal Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> decoding. Furthermore, we show that the Density Evolution analysis of this suboptimal decoder leads to irregular LDPC designs matching the criteria of binary LDPC codes.

A Deviation-Based Conditional Upper Bound on the Error Floor Performance for Min-Sum Decoding of Short LDPC Codes

Conditional upper bounds are given for min-sum decoding of low-density parity-check codes in the error floor region. It is generally thought that absorbing sets, i.e., small collections of variable nodes connected to a relatively small number of unsatisfied check nodes, are the primary source of errors in the error floor region. The conditional upper bounds presented here are based on the assumption that all error floor errors are caused by absorbing sets. In order to bound the probability of error associated with each absorbing set, a directed-edge Tanner graph is used to link absorbing sets to low-weight deviations. These low-weight deviations result when a proportionally large number of nodes within a stopping set belong to an absorbing set contained inside the stopping set. A complete collection of the most problematic absorbing sets, and the minimum deviation weights that result from them, are used to derive a conditional upper bound on the probability of error for min-sum decoding of low-density parity-check codes. Simulation results are given to demonstrate the accuracy of the bound.

Construction of High-Rate Regular Quasi-Cyclic LDPC Codes Based on Cyclic Difference Families

For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to making short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, a new construction method of high-rate regular QC LDPC codes with parity-check matrices consisting of a single row of circulants with the column-weight 3 or 4 is proposed based on special classes of cyclic difference families. The proposed QC LDPC codes can be constructed for various code rates and lengths including the minimum achievable length for a given design rate, which cannot be achieved by the existing high-rate QC LDPC codes. It is observed that the parity-check matrices of the proposed QC LDPC codes have full rank. It is shown that the error correcting performance of the proposed QC LDPC codes of short and moderate lengths is almost the same as that of the existing ones through numerical analysis.

Forced-convergence decoding for LDPC-coded modulation

Low-density parity-check (LDPC)-coded modulation has attracted much attention for its outstanding performance in a bandwidth-limited channel. However, it involves relatively high decoding complexity. Owing to different convergence behaviors of variable nodes, the nodes detected early can be deleted from the Tanner graph, which reduces the size of the parity-check matrix iteration by iteration and thus the decoding complexity. In this paper, we propose a new decoding algorithm for LDPC-coded modulation with reduced complexity. In addition, by deleting the converged bits from the modulation constellation and updating the bit metrics of the demapper with the reduced constellation, more accurate estimation of the bits can be achieved. Simulation results show that, compared with conventional decoding schemes, the proposed method not only greatly reduces the overall decoding complexity for LDPC-coded modulation but also significantly improves decoding performance.

Analysis of the relation between properties of LDPC codes and the tanner graph

A new method for estimating the number of errors guaranteed to be corrected by a low-density parity-check code is proposed. The method is obtained by analyzing edges with special properties of an appropriate Tanner graph. In this paper we consider binary LDPC codes with constituent single-parity-check and Hamming codes and an iterative decoding algorithm. Numerical results obtained for the proposed lower bound exceed similar results for the best previously known lower bounds.

Random Edge-Local Complementation with Applications to Iterative Decoding of High-Density Parity-Check Codes

We describe the application of edge-local complementation (ELC) to a Tanner graph associated with a binary linear code, C. Various properties of ELC are described, mainly the special case of isomorphic ELC operations and the relationship to the automorphism group of the code, Aut(C), and the generalization of ELC to weight-bounding ELC (WB-ELC) operations under which the number of edges remains upper-bounded. ELC generates all systematic parity-check matrices (the orbit) of the code, so WB-ELC facilitates a restriction to low-weight matrices of this orbit. We propose using ELC and WB-ELC as a source of diversity, to improve iterative soft-input soft-output decoding of high-density parity-check (HDPC) codes, with the sum-product algorithm (SPA). A motivation of ELC-enhanced SPA decoding is locality; that diversity is achieved by local graph action, and is well-suited to the local actions that constitute the SPA and allows for parallel software implementation. Simulation data on the error-rate performance of the proposed SPA-ELC and SPA-WBELC iterative decoding algorithms is shown for several HDPC codes. A gain is reported over SPA decoding, and over a recently proposed algorithm to decode HDPC codes using permutations from Aut(C). ELC-enhanced decoding extends the scope of iterative decoding to codes with trivial Aut(C).

Progressive Differences Convolutional Low-Density Parity-Check Codes

We present a new family of low-density parity-check (LDPC) convolutional codes that can be designed using ordered sets of progressive differences. We study their properties and define a subset of codes in this class that have some desirable features, such as fixed minimum distance and Tanner graphs without short cycles. The design approach we propose ensures that these properties are guaranteed independently of the code rate. This makes these codes of interest in many practical applications, particularly when high rate codes are needed for saving bandwidth. We provide some examples of coded transmission schemes exploiting this new class of codes.

Tree-Structure Expectation Propagation for LDPC Decoding Over the BEC

We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). EP generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pair-wise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regime over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law (SL) for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization.

Fountain Coding via Multiplicatively Repeated Non-Binary LDPC Codes

We study fountain codes transmitted over the binary-input symmetric-output channel. For channels with small capacity, receivers in fountain coding systems needs to collects many channel outputs to recover information bits. Since a collected channel output yields a check node in the decoding Tanner graph, the channel with small capacity leads to large decoding complexity. In this paper, we introduce a novel fountain coding scheme with non-binary LDPC codes. The decoding complexity of the proposed fountain code does not depend on the channel. Numerical experiments show that the proposed codes exhibit better performance than conventional fountain codes, especially for moderate number of information bits.

A PEG Construction of Finite-Length LDPC Codes with Low Error Floor

The progressive-edge-growth (PEG) algorithm of Hu et al. is modified to improve the error floor performance of the constructed low-density parity-check (LDPC) codes. To improve the error floor, the original PEG algorithm is equipped with an efficient algorithm to find the dominant elementary trapping sets (ETS's) that are added to the Tanner graph of the under-construction code by the addition of each variable node and its adjacent edges. The aim is to select the edges, among the candidates available at each step of the original PEG algorithm, that prevent the creation of dominant ETS's. The proposed method is applicable to both regular and irregular variable node degree distributions. Simulation results are presented to demonstrate the superior ETS distribution and error floor performance of the constructed codes compared to similar codes constructed by the original and other modifications of the PEG algorithm.

Enhancing the Error-correcting Capability of Imai-Kamiyanagi Codes for Data Storage Systems by Adopting Iterative Decoding using a Parity Check Tree

A novel low-complexity, soft decision technique which allows the decoding of distance-5 double error-correcting Imai-Kamiyanagi codes by using a parity check tree associated with the Tanner graph is proposed. These codes have been applied to memory subsystems and digital storage devices in order to achieve efficient and reliable data processing and storage. For the AWGN channel, gains in excess of 1.5 dB at reasonable bit error rates with respect to conventional hard decision decoding are demonstrated for the (46, 32), (81, 64), and (148, 128) shortened Imai-Kamiyanagi codes.

On the Smallest Absorbing Sets of LDPC Codes From Finite Planes

Absorbing sets, a class of combinatorial structures of the Tanner graph representation of a low-density parity-check (LDPC) code, are known to influence the performance of the code under message passing iterative decoding. In this paper, we study the smallest absorbing sets of LDPC codes constructed from projective planes and Euclidean planes. The lower bounds on the parameters of smallest absorbing sets given by Dolecek are proven to be tight for these two families of LDPC codes. We also analyze the combinatorial properties of the smallest absorbing sets and give conditions necessary and sufficient for a set of bit nodes in the Tanner graph to be a smallest absorbing set. For LDPC codes from projective planes, we further give a condition necessary and sufficient for a smallest absorbing set to be a fully absorbing set. In addition, we show that these smallest absorbing sets are asymptotically not stable, which may explain to some extent the good performance as well as the low error floor expectation of these two families of LDPC codes.