Message-passing Decoding Research Articles

Analysis of Message-Passing Decoding of Finite-Length Concatenated Codes

We analyze the performance of message-passing decoding of finite-length concatenated codes. We first show that the message-passing decoder is closely related to a dual optimization decoder. The connections between these two decoders are further elucidated by proving that both of them attain the same objective function value of a generalized linear programming decoder in the limit as the signal-to-noise ratio (SNR) goes to infinity. Consequently, the framework of pseudo-weight analysis, which was originally proposed for analyzing the linear programming decoder, can be extended to analyze the performance of the message-passing decoder for finite-length codes. We then derive lower bounds to the pseudo-weights of general concatenated codes by utilizing the special structure of their parity-check matrices. We finally present a method to increase the max-fractional weight by adding redundant parity-check constraints and thereby improving the decoding performance. Simulation studies are carried out to assess the performance of the proposed algorithms and substantiate the theoretic claims.

Improve the Performance of LDPC Coded QAM by Selective Bit Mapping in Terrestrial Broadcasting System

In this paper, we employ selective bit mapping to improve the performance of the LDPC coded QAM scheme for terrestrial DTV broadcasting system. The threshold of message-passing decoding can be considerably lowered by selectively mapping the binary components of LDPC codeword to the positions in the m-tuples to be mapped into 2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> QAM symbols. In our approach, the mapping pattern is described by bit-mapping polynomials, based on which density evolution can be applied. The optimization algorithm is developed with two implementation concerns, using the Chinese DTMB standard as an example. Numerical results illustrate that our proposed approach can improve the decoding threshold by 0.05 dB to 0.499 dB depending on the code rate and the order of QAM modulation. Simulation results show that the actual BER improvement varies from 0.09 dB to 0.6 dB with different code-modulation combinations in both single-carrier and OFDM modes.

Exchange of Limits: Why Iterative Decoding Works

We consider communication over binary-input memoryless output-symmetric channels using low-density parity-check codes and message-passing decoding. The asymptotic (in the length) performance of such a combination for a fixed number of iterations is given by density evolution. Letting the number of iterations tend to infinity we get the density evolution (DE) threshold, the largest channel parameter so that the bit error probability tends to zero as a function of the iterations. In practice, we often work with short codes and perform a large number of iterations. It is, therefore, interesting to consider what happens if in the standard analysis we exchange the order in which the blocklength and the number of iterations diverge to infinity. In particular, we can ask whether both limits give the same threshold. Although empirical observations strongly suggest that the exchange of limits is valid for all channel parameters, we limit our discussion to channel parameters below the DE threshold. Specifically, we show that as long as the message reliabilities are bounded and other technical conditions are met, the bit error probability vanishes up to a nontrivial threshold regardless of how the limit is taken. This threshold is equal to the DE threshold when the minimum degree of the variable nodes is at least five and strictly less than the DE threshold for smaller degrees.

Lifting the Fundamental Cone and Enumerating the Pseudocodewords of a Parity-Check Code

The performance of message-passing iterative decoding and linear programming decoding depends on the Tanner graph representation of the code. If the underlying graph contains cycles, then such algorithms could produce a noncodeword output. The study of pseudocodewords aims to explain this noncodeword output. We examine the structure of the pseudocodewords and show that there is a one-to-one correspondence between graph cover pseudocodewords and integer points in a lifted fundamental cone. This gives a simple proof that the generating function of the pseudocodewords for a general parity-check code is rational (a fact first proved by Li, Lu, and Wang (Lecture Notes in Computer Science, vol. 5557, 2009) via other methods). Our approach yields algorithms for producing this generating function and provides tools for studying the irreducible pseudocodewords. Specifically, Barvinok's algorithm and the Barvinok-Woods projection algorithm are applied, and irreducible pseudocodewords are found via a Hilbert basis for the lifted fundamental cone.

A Multimode Shuffled Iterative Decoder Architecture for High-Rate RS-LDPC Codes

For an efficient multimode low-density parity-check (LDPC) decoder, most hardware resources, such as permutators, should be shared among different modes. Although an LDPC code constructed based on a Reed-Solomon (RS) code with two information symbols is not quasi-cyclic, in this paper, we reveal that the structural properties inherent in its parity-check matrix can be adopted in the design of configurable permutators. A partially parallel architecture combined with the proposed permutators is used to mitigate the increase in implementation complexity for the multimode function. The high check-node degree of a high-rate RS-LDPC code leads to challenges in the efficient implementation of a high-throughput decoder. To overcome this difficulty, the variable nodes have been partitioned into several groups, and each group is processed sequentially in order to shorten the critical-path delay and hence increase the maximum operating frequency. In addition, shuffled message-passing decoding is adopted, since fewer iterations can be used to achieve the desired bit-error-rate performance. In order to demonstrate the usefulness of the proposed flexible-permutator-based architecture, one single-mode rate-0.84 decoder and two multimode decoders whose code rates range between 0.79 and 0.93 have been implemented. These decoders can achieve multigigabit-per-second throughput. Using the proposed architecture to support lower rate RS-LDPC codes, e.g., rate-0.568 code, is also investigated.

Interior Point Decoding for Linear Vector Channels Based on Convex Optimization

In the present paper, a novel decoding algorithm for low-density parity-check (LDPC) codes based on convex optimization is presented. The decoding algorithm, which is referred to hereinafter as interior point decoding, is designed for linear vector channels. The linear vector channels include several practically important channels, such as inter-symbol interference channels and partial response (PR) channels. It is shown that the maximum likelihood decoding (MLD) rule for a linear vector channel can be relaxed to a convex optimization problem, which is called a relaxed MLD problem. The proposed decoding algorithm is based on a numerical optimization technique known as the interior point method with barrier functions. Approximate variations of an interior point method based on the gradient descent and Newton methods are used to solve the relaxed MLD problem. Compared with a conventional joint message-passing decoder, from computer simulations, it is observed that the proposed decoding algorithm achieves better BER performance on PR channels with less decoding complexity in several cases. Furthermore, an extension of the proposed algorithm for high-order modulation formats, such as PAM and QAM, is presented.

On the Dynamics of the Error Floor Behavior in (Regular) LDPC Codes

It is shown that dominant trapping sets of regular low-density parity-check (LDPC) codes, so-called absorption sets, undergo a two-phased dynamic behavior in the iterative message-passing decoding algorithm. Using a linear dynamic model for the iteration behavior of these sets, it is shown that they undergo an initial geometric growth phase which stabilizes in a final bit-flipping behavior where the algorithm reaches a fixed point. This analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation. The topology of the dominant absorption sets of an example code, the IEEE 802.3an (2048,1723) regular LDPC code, are identified and tabulated using topological relationships in combination with search algorithms.

Nonbinary LDPC codes constructed based on a cyclic MDS code and a low-complexity nonbinary message-passing decoding algorithm

In this letter, we propose a construction of nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes based on a cyclic maximum distance separable (MDS) code. The parity-check matrices are significantly rank deficient square matrices and their Tanner graphs have a girth of at least 6. The minimum distances of the codes are very respectable as far as LDPC codes are concerned. Based on plurality voting and iterative mechanism, a low-complexity nonbinary massage-passing decoding algorithm is proposed. It only requires finite field operations, integer additions and integer comparisons. Simulation results show that the decoding algorithm is fit for the proposed codes, providing efficient trade-offs between performance and decoding complexity, which suggests that the coding scheme may find some applications in communication or storage systems with high-speed and low-power consumption requirements.

Design of irregular LDPC codes with optimized performance-complexity tradeoff

The optimal performance-complexity tradeoff for error-correcting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the minimization of decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. A sufficient condition is presented for convexity of the complexity measure in the variable edge-degree distribution; when it is not satisfied, numerical experiments nevertheless suggest that the local minimum is unique. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform threshold-optimized codes at long block lengths, within the ensemble of irregular codes.

An implementation-friendly binary LDPC decoding algorithm

We introduce an implementation-friendly binary message-passing decoding method for low-density parity-check (LDPC) codes that does not require the degree information of variable nodes or degree dependent parameters. For hard decision decoding, given its low-complexity, the implementation cost for variable node degree information is an important consideration. We develop an estimation method for the extrinsic error probability (EEP) as well as its analysis. The proposed method offers similar performance as the existing methods for time-invariant decoding in most cases, while it facilitates efficient circuit implementations of the LDPC decoder.

The Manifestation of Stopping Sets and Absorbing Sets as Deviations on the Computation Trees of LDPC Codes

The error mechanisms of iterative message‐passing decoders for low‐density parity‐check codes are studied. A tutorial review is given of the various graphical structures, including trapping sets, stopping sets, and absorbing sets that are frequently used to characterize the errors observed in simulations of iterative decoding of low‐density parity‐check codes. The connections between trapping sets and deviations on computation trees are explored in depth using the notion of problematic trapping sets in order to bridge the experimental and analytic approaches to these error mechanisms. A new iterative algorithm for finding low‐weight problematic trapping sets is presented and shown to be capable of identifying many trapping sets that are frequently observed during iterative decoding of low‐density parity‐check codes on the additive white Gaussian noise channel. Finally, a new method is given for characterizing the weight of deviations that result from problematic trapping sets.

Analysis of Connections Between Pseudocodewords

The role of pseudocodewords in causing non-codeword outputs in linear programming decoding, graph cover decoding, and iterative message-passing decoding is investigated. The three main types of pseudocodewords in the literature-linear programming pseudocodewords, graph cover pseudocodewords, and computation tree pseudocodewords-are reviewed and connections between them are explored. Some discrepancies in the literature on minimal and irreducible pseudocodewords are highlighted and clarified, and the minimal degree cover necessary to realize a pseudocodeword is found. Additionally, some conditions for the existence of connected realizations of graph cover pseudocodewords are given. This allows for further analysis of when graph cover pseudocodewords induce computation tree pseudocodewords. Finally, an example is offered that shows that existing theories on the distinction between graph cover pseudocodewords and computation tree pseudocodewords are incomplete.

Doped fountain coding for minimum delay data collection in circular networks

This paper studies decentralized, Fountain and network-coding based strategies for facilitating data collection in circular wireless sensor networks, which rely on the stochastic diversity of data storage. The goal is to allow for a reduced delay collection by a data collector who accesses the network at a random position and random time. Data dissemination is performed by a set of relays which form a circular route to exchange source packets. The storage nodes within the transmission range of the route's relays linearly combine and store overheard relay transmissions using random decentralized strategies. An intelligent data collector first collects a minimum set of coded packets from a subset of storage nodes in its proximity, which might be sufficient for recovering the original packets and, by using a message-passing decoder, attempts recovering all original source packets from this set. Whenever the decoder stalls, the source packet which restarts decoding is polled/doped from its original source node. The random-walk-based analysis of the decoding/doping process furnishes the collection delay analysis with a prediction on the number of required doped packets. The number of doped packets can be surprisingly small when employed with an Ideal Soliton code degree distribution and, hence, the doping strategy may have the least collection delay when the density of source nodes is sufficiently large. Furthermore, we demonstrate that network coding makes dissemination more efficient at the expense of a larger collection delay. Not surprisingly, a circular network allows for a significantly more (analytically and otherwise) tractable strategies relative to a network whose model is a random geometric graph.

Bounds on the Number of Iterations for Turbo-Like Ensembles Over the Binary Erasure Channel

This paper provides simple lower bounds on the number of iterations which is required for successful message-passing decoding of some important families of graph-based code ensembles (including low-density parity-check (LDPC) codes and variations of repeat-accumulate codes). The transmission of the code ensembles is assumed to take place over a binary erasure channel, and the bounds refer to the asymptotic case where we let the block length tend to infinity. The simplicity of the bounds derived in this paper stems from the fact that they are easily evaluated and are expressed in terms of some basic parameters of the ensemble which include the fraction of degree-2 variable nodes, the target bit erasure probability, and the gap between the channel capacity and the design rate of the ensemble. This paper demonstrates that the number of iterations which is required for successful message-passing decoding scales at least like the inverse of the gap (in rate) to capacity, provided that the fraction of degree-2 variable nodes of these turbo-like ensembles does not vanish (hence, the number of iterations becomes unbounded as the gap to capacity vanishes).

Exact thresholds for low-density parity-check codes over the binary erasure channel

A simple method for determining the threshold of irregular LDPC codes over the binary erasure channel (BEC) under message-passing decoding is proposed. An exact formula for calculating the threshold of irregular LDPC codes over the BEC is proved. This generalizes the known result on the threshold of regular LDPC codes to irregular LDPC codes. Moreover, our new method can avoid the computation of the inverse of the degree distribution function for irregular LDPC codes. Numerical results demonstrate the correctness of our proposed method.

Burst erasure correcting LDPC codes

In this paper low-density parity-check (LDPC) codes are designed for burst erasure channels. Firstly, lower bounds for the maximum length erasure burst that can always be corrected with message-passing decoding are derived as a function of the parity-check matrix properties. We then show how parity-check matrices for burst erasure correcting LDPC codes can be constructed using superposition, where the burst erasure correcting performance of the resulting codes is derived as a property of the stopping set size of the base matrices and the choice of permutation matrices for the superposition. This result is then used to design both single burst erasure correcting LDPC codes which are also resilient to the presence of random erasures in the received bits and LDPC codes which can correct multiple erasure bursts in the same codeword.

Message-passing iterative decoding between detector and RSC code decoder for PMR channel

For higher density perpendicular magnetic recording (PMR) systems, it is difficult to expect an improvement in performance when only partial response maximum likelihood (PRML) or noise predictive maximum likelihood (NPML) detection is used. Hence, we consider a system that performs better than either PRML or NPML. We concatenate a channel detector using a recursive systematic convolutional (RSC) code decoder and a message-passing algorithm. The RSC code itself is simple but its performance is inferior to other iterative decoding methods. Therefore, we used channel iteration between the channel detector and the RSC decoder in order to improve performance. We also used a precoding and random interleaver. We evaluated the performance on the PMR channel with jitter noise.

Sequential message-passing decoding of LDPC codes by partitioning check nodes

In this paper, we analyze the sequential message- passing decoding algorithm of low-density parity-check (LDPC) codes by partitioning check nodes. This decoding algorithm shows better bit error rate (BER) performance than the conventional message-passing decoding algorithm, especially for the small number of iterations. Analytical results indicate that as the number of partitioned subsets of check nodes increases, the BER performance is improved. We also derive the recursive equations for mean values of messages at check and variable nodes by using density evolution with a Gaussian approximation. From these equations, the mean values are obtained at each iteration of the sequential decoding algorithm and the corresponding BER values are calculated. They show that the sequential decoding algorithm converges faster than the conventional one. Finally, the analytical results are confirmed by the simulation results.

Decoding with Early Termination for Raptor Codes

Rateless codes, and especially Raptor codes, have received considerable attention in the recent past due to their inherent ability to adapt to channel conditions and their capacity- approaching performance. Since decoding of rateless codes typically involves multiple decoding attempts, early termination of such attempts is mandatory for overall efficient decoding. In this letter, we propose a new decoding scheme with early termination that is particularly suited for rateless codes. Simulation results for the example of the binary symmetric channel show complexity reductions (in terms of the total required number of decoding iterations) by 87% compared to conventional message-passing decoding and 54% compared to a recently proposed incremental decoding scheme for Raptor codes.

Reliability-Based Hybrid ARQ Scheme with Encoded Parity Bit Retransmissions and Message Passing Decoding

Reliability-based hybrid ARQ (RB-HARQ) is a kind of incremental-redundancy ARQ recently introduced. In the RB-HARQ, the receiver returns both NAK signal and set of unreliable bit indices if the received sequence is not accepted. Since each unreliable bit index is determined by the bitwise posterior probability, better approximation of that probability becomes crucial as the number of retransmissions increases. Assuming the systematic code for the initial transmission, the proposed RB-HARQ scheme has the following two features: (a) the sender retransmits newly encoded and interleaved parity bits corresponding to the unreliable information bits; (b) the retransmitted parity bits as well as the initial received sequence are put all together to perform the message passing decoding i.e. the suboptimal MAP decoding. Finally, simulation results are shown to evaluate the above two features.