Belief Propagation Decoding Research Articles

Stochastic Bit-Wise Iterative Decoding of Polar Codes

Polar codes have received recent attention due to their potential to be applied in advanced wireless communication protocols such as the fifth generation mobile communication system (5G). Among the existing decoding algorithms, Belief Propagation (BP) exhibits high-throughput, low-latency, and soft output with a high hardware cost. Stochastic computing, as a form of approximate computing, provides a potential low-cost implementation solution for the BP algorithm. However, existing stochastic BP decoders suffer from a relatively long decoding latency resulting in low hardware efficiency. In this paper, a novel bit-wise iterative stochastic decoding architecture for the BP algorithm is proposed to improve the throughput and hardware efficiency. By utilizing the frozen bits of polar codes and stochastic computing, multiple novel optimization methods are presented to further speed up convergence and increase the hardware efficiency.

Systematic Fountain Codes for Massive Storage Using the Truncated Poisson Distribution

Erasure codes for distributed storage systems (DSS) are required to offer systematic encoding, low repair locality, low encoding/decoding complexity, and low decoding/storage overhead. However, the information theoretical bounds have shown that all these metrics might need to be carefully traded-off with one another. In this paper, we consider systematic Fountain codes with belief propagation (BP) decoding for massive scale DSSs. Analyzing the role of some degrees in the BP decoding process, we propose using the truncated Poisson distribution (TPD) for encoded symbol degrees. Identifying encoded symbol redundancy as a factor that degrades decoding overhead, we derive the probability of redundancy during the BP decoding process and use this as a tool for determining the Poisson parameter. Our proposed solution nicely addresses the first three DSS metrics with a slight toll on storage/decoding overhead. Through simulations, we show that the decoding overhead performance of the proposed scheme exhibits significant improvement over some existing Fountain code degree distributions in the literature. For instance, at a decoding overhead of 60%, we achieve a data loss probability closely approaching 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> , while other Fountain codes compared are about a factor of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> higher.

More Accurate Analysis of Sum-Product Decoding of LDPC Codes Using a Gaussian Approximation

This letter presents a more accurate mathematical analysis, with respect to the one performed in Chung et al.'s 2001 paper, of belief-propagation decoding for low-density parity-check (LDPC) codes on memoryless binary input-additive white Gaussian noise channels, when considering a Gaussian approximation (GA) for message densities under density evolution. The recurrent sequence, defined in Chung et al.'s 2001 paper, describing the message passing between variable and check nodes follows from the GA approach and involves the function φ(x), therein defined, and its inverse. The analysis of this function is here resumed and studied in depth to obtain tighter upper and lower bounds on it. Moreover, unlike the upper bound given in the above cited paper, the tighter upper bound on φ(x) is invertible. This allows a more accurate evaluation of the asymptotical performance of sum-product decoding of LDPC codes when a GA is assumed.

Belief Propagation Bit-Flip Decoder for Polar Codes

The bit-flip method has been successfully applied to the successive cancellation (SC) decoder to improve the block error rate (BLER) performance for polar codes in the finite code length region. However, due to the sequential decoding, the SC decoder inherently suffers from longer decoding latency than that of the belief propagation (BP) decoder with efficient early stopping criterion. It is natural to ask how to perform bit-flip in a polar BP decoder. In this paper, bit-flip is introduced into the BP decoder for polar codes. The idea of critical set (CS), that is, originally proposed by Zhang et al. for identifying unreliable bits in a SC bit-flip decoder, is extended to the BP decoder here. After revealing the relationship between CS and the incorrect BP decoding results, critical set with order ω (CS-ω) is constructed to identify unreliable bit decisions in polar BP decoding. The simulation results demonstrate that compared with the conventional BP decoder, the BLER of the proposed bit-flip decoder can achieve significant signal-to-noise ratio (SNR) gain which is comparable to that of a cyclic redundancy check-aided SC list decoder with a moderate list size. In addition, the decoding latency of the proposed BP bit-flip decoder is only slightly higher than that of the conventional BP decoder in the medium and high SNR regions.

Low Complexity Polar Decoder for 5G Embb Control Channel

Polar codes have become the channel coding scheme for control channel of enhanced mobile broadband in the fifth generation (5G) communication systems. Belief propagation (BP) decoding of polar codes has the advantage of low decoding latency and high parallelism but suffers from high complexity. In this paper, a low complexity BP decoder is proposed for polar codes. We reduce the computational complexity by two steps. First, the cyclic redundancy check is concatenated to the decoder in order to decrease the number iterations of the BP algorithm. Then, a threshold is proposed based on Gaussian approximation to save the computational complexity of BP nodes. If the log-likelihood ratio of a node in the tanner graph is larger than the threshold, this node is no longer updated during the rest of the decoding process. The simulation results show that the proposed scheme has a similar block error rate performance with the original BP decoder, while the computational complexity is reduced significantly.

Mapping Design for $2^{M}$ -Ary Bit-Interleaved Polar Coded Modulation

This paper proposes a mapping design for bit-interleaved polar coded modulation (BIPCM) systems with belief propagation (BP) decoding. We first introduce a two-layer bipartite graph to represent BIPCM, where a new mapping graph linking polar graph to modulator is added to the conventional factor graph. Then, a mapping design is proposed and the design paradigm is to separate sub-channels with lower reliability to different stopping trees of polar codes, aiming to make sure that each stopping tree receives reliable extrinsic information from demodulator. The proposed mapping algorithm is employed for BIPCM with traditional polar codes over 16-quadrature amplitude modulation (QAM) and 256-QAM. Numerical results show that our scheme can improve the error-correcting performance compared to the conventional scheme with a random mapping. Furthermore, to meet code-length requirement of different modulation orders, we propose an efficient method to construct flexible-length polar code (FLPC) by coupling several short length polar codes with a repeat-accumulate (RA) code. Also, the proposed FLPC is employed in the BIPCM system, with the designed mapping algorithm, simulation result also reveals that the block error rate performance of proposed BIPCM scheme with BP decoding outperforms the one with successive cancellation decoding by providing a gain up to 1 dB.

An implementation of belief propagation decoder with combinational logic reduced for polar codes

An implementation of belief propagation decoder with combinational logic reduced for polar codes

Information-Bottleneck Decoding of High-Rate Irregular LDPC Codes for Optical Communication Using Message Alignment

In high-throughput applications, low-complexity and low-latency channel decoders are inevitable. Hence, for low-density parity-check (LDPC) codes, message passing decoding has to be implemented with coarse quantization—that is, the exchanged beliefs are quantized with a small number of bits. This can result in a significant performance degradation with respect to decoding with high-precision messages. Recently, so-called information-bottleneck decoders were proposed which leverage a machine learning framework (i.e., the information bottleneck method) to design coarse-precision decoders with error-correction performance close to high-precision belief-propagation decoding. In these decoders, all conventional arithmetic operations are replaced by look-up operations. Irregular LDPC codes for next-generation fiber optical communication systems are characterized by high code rates and large maximum node degrees. Consequently, the implementation complexity is mainly influenced by the memory required to store the look-up tables. In this paper, we show that the complexity of information-bottleneck decoders remains manageable for irregular LDPC codes if our proposed construction approach is deployed. Furthermore, we reveal that in order to design information bottleneck decoders for arbitrary degree distributions, an intermediate construction step which we call message alignment has to be included. Exemplary numerical simulations show that incorporating message alignment in the construction yields a 4-bit information bottleneck decoder which performs only 0.15 dB worse than a double-precision belief propagation decoder and outperforms a min-sum decoder.

Augmented decoders for LDPC codes

The performance of a belief propagation decoder for low-density parity-check codes is limited by the presence of trapping sets in the code’s graph. This leads to an error floor at higher signal-to-noise ratios. We propose the use of an augmented decoder which operates by iteratively decoding on a set of graphs which have a subset of repeated check nodes. We compare the augmented decoder to other modified belief propagation decoders that have been presented in the literature. We show that for all the codes considered, the augmented decoder yields the best frame error rate in the error floor region.

Comments on “Edge-Based Dynamic Scheduling for Belief-Propagation Decoding of LDPC Codes”

In this paper, we provide comments on the recent paper by Aslam et al. that proposed two low-complexity edge-based scheduling schemes for belief-propagation (BP) decoding of low-density parity-check (LDPC) codes. In their schemes, the check-node reliability threshold (τ) is important to select the updating edges. However, it is found that the selected value of τ in the above literature is not optimal. In this comment, we present a method that leads to a better choice of τ.

Low‐complexity bound on irregular LDPC belief‐propagation decoding thresholds using a Gaussian approximation

Since irregular low-density parity-check (LDPC) codes are known to perform better than regular ones, and to exhibit, like them, the so-called ‘threshold phenomenon’, this Letter investigates a low-complexity upper bound on belief-propagation decoding thresholds for this class of codes on memoryless binary input additive white Gaussian noise channels, with sum-product decoding. A simplified analysis of the belief-propagation decoding algorithm is used, i.e. consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes.

Belief Propagation List Decoding of Polar Codes

We propose a belief propagation list (BPL) decoder with comparable performance to the successive cancellation list (SCL) decoder of polar codes, which already achieves the maximum likelihood (ML) bound of polar codes for sufficiently large list size $L$. The proposed decoder is composed of multiple parallel independent belief propagation (BP) decoders based on differently permuted polar code factor graphs. A list of possible transmitted codewords is generated and the one closest to the received vector, in terms of Euclidean distance, is picked. To the best of our knowledge, the proposed BPL decoder provides the best performance of plain polar codes under iterative decoding known so far. The proposed algorithm does not require any changes in the polar code structure itself, rendering the BPL into an alternative to the SCL decoder, equipped with a soft output capability enabling, e.g., iterative detection and decoding to further improve performance. Further benefits are lower decoding latency compared to the SCL decoder and the possibility of high throughput implementations. Additionally, we show that a different selection strategy of frozen bit positions can further enhance the error-rate performance of the proposed decoder.

Knowledge‐aided iterative detection and decoding for multiuser multiple‐antenna systems

In this work, the authors assess the performance and latency of a receiver design for multiuser multiple-antenna systems using low-density parity-check (LDPC) codes with iterative detection and decoding (IDD). The proposed knowledge-aided IDD (KA-IDD) receiver employs a parallel interference cancellation detector with refined iterative processing and a reweighted belief propagation (BP) decoder. Additionally, two BP-based decoding algorithms are proposed for improved performance of the decoder. The first cycle knowledge-aided reweighted-BP decoder takes advantage of the information about the distribution of cycles in the Tanner graph and the second expansion knowledge-aided reweighted-BP decoder expands the original graph into a set of subgraphs, then locally optimises the reweighting parameters of each subgraph. Novel reweighting parameter optimisation strategies are developed for decoding regular or irregular LDPC codes. The developed schemes optimise the parameters off-line and neither of these proposed algorithms imposes extra computational complexity to the on-line decoding procedure. Furthermore, the proposed schemes reduce the decoding latency. Simulation results show that the proposed KA-IDD scheme and algorithms outperform prior art and require a reduced number of decoding iterations.

Scaling factor optimization of turbo-polar iterative decoding

In this paper, we investigate the weighted iterative decoding to improve the performance of turbo-polar code. First of all, a minimum weighted mean square error criterion is proposed to optimize the scaling factors (SFs). Secondly, for two typical iterative algorithms, such as soft cancellation (SCAN) and belief propagation (BP) decoding, genie-aided decoders are proposed as the ideal reference of the practical decoding. Guided by this optimization framework, the optimal SFs of SCAN or BP decoders are obtained. The bit error rate performance of turbo-polar code with the optimal SFs can achieve 0.3 dB or 0.7 dB performance gains over the standard SCAN or BP decoding respectively.

An Improved Belief Propagation Decoding of Concatenated Polar Codes With Bit Mapping

Bit-channels of finite-length polar codes are not fully polarized, and the negative influences from semi-polarized channels are not negligible. In this letter, we consider the concatenated system of an inner polar code and an outer low-density parity-check (LDPC) code to improve the error correction performance of finite-length polar code with belief propagation (BP) decoding. We propose a bit mapping method between LDPC coded bits and polarized channels, aiming to match the coding protection of LDPC codes with polarization of polar codes. Simulation results show that, the concatenated polar codes with bit mapping outperform the pure polar codes with BP decoding and concatenated polar codes with traditional consecutive bit mapping.

A low-latency list decoder for polar codes

Successive cancellation (SC) is a low complexity serial decoding algorithm for polar codes, and successive cancellation list (SCL) can achieve excellent error-correcting performance. However, SCL decoder suffers from long decoding latency compared with belief propagation (BP) decoder. In this paper, a low-latency list decoder whose latency performance can approach that of BP deocder is proposed. A prunable subtree recognizing scheme based on ${\boldsymbol~H}$-Matrix check is proposed by taking the reliability of frozen bits into account. Then, a latency-reduced list decoder based on the prunable constituent codes is proposed. Simulation results show that the decoding latency of proposed list scheme can be reduced significantly, especially for high signal noise ratio (SNR) region.

Finite-Length Analysis of Spatially-Coupled Regular LDPC Ensembles on Burst-Erasure Channels

Regular spatially-coupled low-density parity-check ensembles have gained significant interest, since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this paper, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per code word and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provide new tight lower bounds for the block erasure probability (PB) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. We further show that expurgating the ensemble can improve the block erasure probability by several orders of magnitude. Later we extend our methodology to more general channel models. In a first extension, we consider bursts that can start at a random location in the code word and span across multiple spatial positions. Besides the finite length analysis, we determine by means of density evolution the maximum correctable burst length. In a second extension, we consider the case where in addition to a single burst, random bit erasures may occur. Finally, we consider a block erasure channel model which erases each spatial position independently with some probability p, potentially introducing multiple bursts simultaneously. All results are verified using Monte-Carlo simulations.

A joint-LDPC decoding scheme based on retention error characteristics for MLC NAND flash memory

With the aggressive scaling and the Multi-Level Cell (MLC) technology, the reliability of NAND flash degrades seriously. Therefore, the traditional error correction schemes are becoming less and less effective. Low-Density Parity-Check (LDPC) code and soft-decision memory sensing will be more suitable in the next generation Error Correction Code (ECC) technologies for MLC NAND flash memory. However, LDPC ECC schemes have their own bottleneck due to the unacceptably long read latency during the decoding scheme. The read latency primarily includes two aspects: repeated read cycles and iterative belief-propagation decoding. Motivated by the precise probability information can improve the performance of LDPC decoding, we propose a novel Joint-LDPC (J-LDPC) scheme to reduce the decoding latency of LDPC ECC. The key idea of the proposed J-LDPC scheme is to introduce a new calculating method of probability information in a pair of shared flash pages, according to the retention error characteristics of MLC NAND flash rather than several reading operations. Through a large amount of experiments from our own hardware-software co-designed experimental platform, compared with the traditional soft-decision LDPC scheme, the proposed J-LDPC scheme can reduce the average iterations of LDPC decoder by 50% and enhance the error correction strength by 58%.

Quad-level cell NAND design and soft-bit generation for low-density parity-check decoding in system-level application

QLC (Quad-Level Cell) NAND flash will be one of the future technologies for next generation memory chip after three-dimensional (3D) TLC (Triple-Level Cell) stacked NAND flash. In QLC device, data errors will easily occur because of 24 data levels in the limited voltage range. This paper studies QLC NAND technology which is 4 bits per cell. QLC programming methods based on 16 voltage levels and reading method based on “half-change” Gray coding are researched. Because of the probable error impact of QLC NAND cell’s voltage change, the solution of generating the soft information after XOR (exclusive OR) the soft bits by internal read mechanism is presented for Low-Density Parity-Check (LDPC) Belief Propagation (BP) decoding in QLC design for its system level application.

Rateless Multiple Access: Asymptotic Throughput Analysis and Improvement With Spatial Coupling

A novel random code-domain non-orthogonal multiple access (NOMA) framework, rateless multiple access (RMA), is studied in this paper. In RMA, instead of granting each user specific radio resources in a fixed and centralized manner, the access point simply assigns to it a random access control function , according to which the user independently chooses a pseudo-random number of symbols each time and transmits their weighted sum until receiving an acknowledgment that indicates the successful recovery of its information. In this way, the transmission process of each individual user as well as that of all the users as a whole, resembles a special linear superposition rateless encoder and, thus, information can be retrieved with the low-complexity belief propagation (BP) algorithm at the access point. In this paper, the asymptotic throughput of RMA with BP decoding is analyzed. Then, in order to bridge the performance gap between BP decoding and the optimal MAP (maximum a posteriori ) decoding, we apply the most recently developed coding technique of spatial coupling to RMA and propose an enhanced version, SC-RMA, in which all the user codewords are properly spatially-coupled across different subsets of REs. We prove that SC-RMA asymptotically approaches the channel sum-rate capacity with high adaptability and low-signaling overhead, which makes it a viable candidate for massive access.