Iterative Belief Propagation Decoding Research Articles

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%.

Polar Code Constructions Based on LLR Evolution

Polar code constructions based on mutual information or Bhattacharyya parameters of bit-channels are intended for hard-output successive cancellation (SC) decoders, and thus might not be well designed for use with other decoders, such as soft-output belief propagation (BP) decoders or successive cancellation list (SCL) decoders. In this letter, we use the evolution of messages, i.e., log-likelihood ratios, of unfrozen bits during iterative BP decoding of polar codes to identify weak bit-channels, and then modify the conventional polar code construction by swapping these bit-channels with strong frozen bit-channels. The modified codes show improved performance not only under BP decoding, but also under SCL decoding. The code modification is shown to reduce the number of low-weight codewords, with and without CRC concatenation.

Decoding Binary Linear Block Codes Using Local Search

This paper presents a novel iterative hard decision decoding algorithm for binary linear block codes over a binary symmetric channel (BSC). The problem is formulated as a 0-1 integer programming problem which is known to be NP-hard. When the crossover probability c of the channel is known, the solution space of the decoding problem can be decreased to a sphere whose radius is related to c. Using the penalty function method, the problem is reformulated on this reduced solution space. Then an iterative multi-flip local search algorithm is designed to find the global solution of this decoding problem. For a code with minimum distance d, when the radius of the sphere is not greater than d-1/2, this algorithm has the maximum likelihood (ML) certificate property, i.e., if the decoder outputs a codeword, it is guaranteed to be the ML codeword. Compared to the probabilistic suboptimal iterative belief propagation (BP) decoder, this approach has lower complexity and better performance. Numerical results show that in terms of speed and performance the proposed decoding method outperforms BP decoding in the error floor region.

Window-constrained interconnect-efficient progressive edge growth LDPC codes

One of the attractive features of low-density parity-check (LDPC) codes is the parallel iterative nature of their iterative belief propagation decoding, making them amenable to efficient hardware implementation. However, for an arbitrary code construction, the random-like connections between the code's Tanner graph variable and check nodes makes fully-parallel implementation a difficult task as this leads to complex interconnect wiring and routing congestion. In this paper, we present a novel LDPC code design approach, based on the progressive edge growth (PEG) Tanner graph construction, to solve the problem of dense connections between processing nodes. The approach is based on controlling the maximum connection length between processing nodes in order to make fully parallel implementation feasible. The proposed algorithm offers a good compromise between error correction performance and decoder complexity. Simulation results and FPGA-based implementation comparisons are presented to demonstrate the advantages of the proposed LDPC code constructions, and it is shown that, with proper window-constrained node placement design, an improvement of up to 40% in interconnect efficiency is achievable without any significant degradation in error correction capability.

New stopping criteria for iterative decoding of LDPC codes in H‐ARQ systems

SUMMARYUnder severely unreliable channel, decoding of error‐correcting codes frequently fails, which requires a lot of computational complexity, especially, in the iterative decoding algorithm. In hybrid automatic repeat request systems, most of computation power is wasted on failed decoding if a codeword is retransmitted many times. Therefore, early stopping of iterative decoding needs to be adopted. In this paper, we propose a new stopping algorithm of iterative belief propagation decoding for low‐density parity‐check codes, which is effective on both high and low signal‐to‐noise ratio ranges and scalable to variable code rate and length. The proposed stopping algorithm combines several good stopping criteria. Each criterion is extremely simple and will not be a burden to the overall system. With the proposed stopping algorithm, it is shown via numerical analysis that the decoding complexity of hybrid automatic repeat request system with adaptive modulation and coding scheme can be fairly reduced. Copyright © 2012 John Wiley & Sons, Ltd.

Lossy Multicasting Over Binary Symmetric Broadcast Channels

Lossy multicasting of a set of independent, discrete-time, continuous-amplitude source components under the mean square error distortion measure over binary symmetric broadcast channels is investigated. The practically appealing concatenation of successive refinement source coding with broadcast coding and, specifically, time-sharing of linear binary codes, is considered. Three different system optimization criteria are formulated for the lossy multicasting problem. The resulting system optimization is fairly general and applies to a variety of combinations of successive refinement source codes and channel codes. The system optimization is investigated in depth for a class of channel optimized quantization with successive refinement, obtained by using standard embedded scalar quantizers and linear mapping of the (redundant) quantizer bitplanes onto channel codewords by using a systematic Raptor encoder. This scheme is referred to as quantization with linear index coding (QLIC). Unlike existing literature on progressive transmission with unequal error protection or channel optimized quantization, the focus here is on the regime of moderate-to-large code block length and the power of modern sparse-graph codes with iterative belief propagation decoding is leveraged. In this regime, the system optimization takes on the form of simple convex programming that reduces to linear programming for QLIC. The performance of QLIC compares favorably with respect to the state of the art channel optimized quantization in the conventional setting of a single Gaussian source over a binary symmetric channel. For the multicast scenario, the performance gap incurred by the practical QLIC design with respect to ideal source and channel codes is quantified.

Iterative Decoding Threshold Analysis for LDPC Convolutional Codes

An iterative decoding threshold analysis for terminated regular LDPC convolutional (LDPCC) codes is presented. Using density evolution techniques, the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the AWGN channel with binary inputs. It is shown that for a terminated LDPCC code ensemble, the thresholds are better than for corresponding regular and irregular LDPC block codes.

Error performances of multi shift-register convolutional self-doubly-orthogonal codes

A class of orthogonal convolutional codes using a multi shift-register encoder and featuring self-doubly-orthogonal properties is analyzed under iterative decoding. The lower bounds of error performances of these codes can be approached within typically three to five iterations at moderate signal-to-noise ratios using either iterative threshold (TH) decoding or belief propagation (BP) decoding. Compared with iterative BP decoding, it is shown that iterative threshold decoding for these codes yields a much lower complexity at the same decoding latency.

Tradeoff of complexity and latency of iterative decoding for orthogonal convolutional codes

The complexities and latencies of iterative belief propagation (BP) and threshold decoding techniques for a new class of convolutional self-orthogonal codes are analysed using forward-only pipeline architectures. Computer simulation results show that over the additive white Gaussian noise channel at moderate signal-to-noise ratios, BP decoding yields one half the decoding delay when compared with threshold decoding at essentially the same error performances. However, the reduction in the decoding delay using BP is obtained at the cost of an increase of the implementation complexity.

Turbo Decoding on the Binary Erasure Channel: Finite-Length Analysis and Turbo Stopping Sets

This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stopping sets</i> . We describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum-size <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">turbo stopping set</i> which is also a subset of the set of erased positions. Furthermore, we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results. Finally, we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption, and an efficient enumeration algorithm of small-size turbo stopping sets for a particular interleaver. The solution is based on the algorithm proposed by Garello <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2001 to compute an exhaustive list of all low-weight codewords in a turbo code.

Incremental Redundancy Hybrid ARQ Schemes Based on Low-Density Parity-Check Codes

We study the throughput of hybrid automatic retransmission request (H-ARQ) schemes based on incremental redundancy (IR) over a block-fading channel. We provide an information-theoretic analysis assuming binary random coding and typical-set decoding. Then, we study the performance of low-density parity-check (LDPC) code ensembles with iterative belief-propagation decoding, and show that, under the hypothesis of infinite-length codes, LDPCs yield almost optimal performance. Unfortunately, standard finite-length LDPC ensembles incur a considerable performance loss with respect to their infinite-length counterpart, because of their poor frame-error rate (FER) performance. In order to recover part of this loss, we propose two simple yet effective methods: using a modified LDPC ensemble designed to improve the FER; and using an outer selective-repeat protocol acting on smaller packets of information bits. Surprisingly, these apparently very different methods yield almost the same performance gain and recover a considerable fraction of the optimal throughput, thus making practical finite-length LDPC codes very attractive for data wireless communications based on IR H-ARQ schemes.