Spatially Coupled Low-density Parity-check Research Articles

Window code parameters of spatially-coupled LDPC codes

In this paper, we define a window code to be the portion of a Spatially-coupled low-density parity check (SC-LDPC) code seen by a single iteration of a windowed decoder. We consider the design of SC-LDPC codes for windowed decoding via optimization of the window code. In particular, because iterative decoding is optimal on codes with cycle-free graph representations, we ask fundamental questions about the construction and parameters of cycle-free window codes. We show that it is possible to have an SC-LDPC code with cycles and with cycle-free window codes. We consider the relationship between the distance of the window code and the distance of the SC-LDPC code. Further, we show that SC-LDPC codes with MDS window codes exist, and all such codes are asymptotically bad. This work gives insight into the tradeoffs between window code parameters and performance of the SC-LDPC code.

Effective bi-directional overlapped sliding window decoding of SC-LDPC codes

In this paper, a bi-directional sliding window decoder is proposed for spatially coupled low-density parity-check (SC-LDPC) codes, which improves the decoding complexity and performance compared to the conventional sliding window decoding (SWD) by sharing messages at the overlapped part of forward and backward decoding windows. Moreover, by using proper scaling factors that determine the weight of each message at the overlapped part of two sliding windows, good local decoding effects can be efficiently spread out to both ends of SC-LDPC code during decoding process. Such effective message updates of the proposed bi-directional overlapped sliding window decoding (BO-SWD) improve error floor performance compared to the conventional SWD. The validity of BO-SWD is verified by simulation with various SC-LDPC ensembles.

Finite-Length Analysis for Spatially Coupled LDPC Codes Based on Base Matrix.

Spatially coupled low density parity check (SC-LDPC) are prominent candidates for future communication standards due to their "threshold saturation" properties. To evaluate the finite-length performance of SC-LDPC codes, a general and efficient finite-length analysis from the perspective of the base matrix is proposed. We analyze the evolution of the residual graphs resulting at each iteration during the decoding process based on the base matrix and then derive the expression for the error probability. To verify the effectiveness of the proposed finite-length analysis, we consider the SC-LDPC code ensembles constructed by parallelly connecting multiple chains (PC-MSC-LDPC). The analysis results show that the predicted error probabilities obtained by using the derived expression for the error probability match the simulated error probabilities. The proposed finite-length analysis provides a useful engineering tool for practical SC-LDPC code design and for analyzing the effects of the code parameters on the performances.

Block Markov superposition transmission of LDPC codes

In this paper, based on the block Markov superposition transmission (BMST) technique, we present a new class of coupled low-density parity-check (LDPC) codes for the transport block (TB)-based transmission to improve the error-correcting performance. For encoding, the previous LDPC codewords corresponding to a TB (at prior time slot) are interleaved and superimposed onto the current LDPC codewords, resulting in the transmitted codewords. For decoding, the sliding window decoding algorithm with sum-product or min-sum implementations can be employed, inheriting a relatively low-latency decoding. A distinguished advantage of the proposed coded transmission over spatially coupled LDPC (SC-LDPC) codes is that the encoder/decoder of the proposed codes can be designed by reusing the encoder/decoder architecture of component block LDPC codes. To analyze the waterfall performance of BMST-LDPC code ensembles, we present the protograph-based EXIT chart analysis, which can efficiently predict the error-correcting performance in waterfall region. To analyze the error-floor performance of BMST-LDPC codes, we employ the genie-aided (GA) lower bound, which can efficiently predict the error-correcting performance in error-floor region. For ease of implementation, the BMST-LDPC codes are constructed by taking the (2, 4)-raptor-like LDPC codes or the 5G LDPC codes as the basic components. The numerical results reveal that the proposed codes can have capacity-approaching performance, exhibiting a gap of 0.007 ​dB away from the corresponding Shannon limit. They also reveal that, by using the proposed BMST construction, the error-correcting performance of the original 5G block LDPC codes can be significantly improved, achieving coding gains up to one dB over the AWGN channels and two dB over the fast fading channels.

Free-Ride Coding for Constructions of Coupled LDPC Codes

Free-ride coding, as an approach that admits transmission of a few extra bits over a low-density parity-check (LDPC) coded link without bandwidth expansion, is applied in this paper to construct coupled LDPC codes. Firstly, we present a syndrome channel model and derive the lower and upper bounds on its capacity (referred to as accessible capacity), indicating the feasibility of the reliable transmission of extra bits. Secondly, we present the performance evaluation on both the word error rate (WER) and the bit error rate (BER) for the free-ride codes with simple lower and upper bounds. Then we propose three applications of free-ride coding to construct coupled LDPC codes, including implicit globally-coupled LDPC (GC-LDPC) codes, partial product-LDPC codes, and terminated spatially-coupled LDPC (SC-LDPC) codes, all of which have the figure of merits that they share the same code rates with the basic component LDPC codes. Simulation results show that: 1) the proposed GC-LDPC codes can outperform the component LDPC codes, yielding a coding gain of up to 0.8 dB; 2) the proposed product codes with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(3,6)$ </tex-math></inline-formula> -regular LDPC component codes of length 1024 can lower the WER from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-2}$ </tex-math></inline-formula> down to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-6}$ </tex-math></inline-formula> at the SNR around 2 dB; 3) the proposed terminated SC-LDPC codes can perform as well as the conventional terminated SC-LDPC codes but without any rate loss.

Windowed-based synchronization error-correction for spatially coupled codes

This letter proposes a windowed-based error correction scheme for spatially coupled low-density parity-check (SC-LDPC) codes over channels with insertion and deletion errors. We first introduce the windowed detection and decoding scheme and then examine its asymptotic performance with several types of SC-LDPC codes. Subsequently, we demonstrate that a significant performance gain with lower latency and delay can be obtained using finite-length SC-LDPC codes.

Beyond 100 Gbit/s Pipeline Decoders for Spatially Coupled LDPC Codes

Low-density parity-check (LDPC) codes are a well-established class of forward error correction codes that provide excellent error correction performance for large code block sizes. However, for throughputs toward 1 Tbit/s, as expected for B5G systems, state-of-the-art LDPC soft decoders are restricted to short code block sizes of several hundred to thousands of bits due to routing congestion challenges, limiting the overall communications performance of the transmission system. Spatially coupled LDPC (SC-LDPC) codes and respective sliding window decoding methods show the potential to overcome these block size restrictions. However, in contrast to conventional LDPC codes little literature exists on the efficient hardware implementation of respective high-throughput decoders. In this work, we present the first in-depth investigation on the implementation of SC-LDPC decoders for throughputs beyond 100 Gbit/s. For an N=51328, R=0.8 terminated SC-LDPC code with sub-block size c=640 and coupling width m_s=1, we explore various design trade-offs, including row- and column-wise decoding, non-overlapping and overlapping window scheduling, and processor pipelining. To the best of our knowledge, this is the first description of a column-wise SC-LDPC decoding architecture in the literature. We complement the algorithmic investigation with the virtual silicon implementation of all presented decoders in a 22nm FD-SOI technology.

Integer programming‐based non‐uniform window decoding schedules for spatially coupled low‐density parity‐check codes

Spatially coupled low-density parity-check (SC-LDPC) codes generally use a window decoding scheme, which is known to yield a near-optimal decoding, compared to full block decoding. Recently, a non-uniform schedule has been proposed to eliminate unnecessary updates of variable nodes within a window: this schedule is generated based on the behaviour of variable node updates analysed by density evolution. Here, the authors present a new non-uniform schedule based on integer programming, whereby the objective functions and constraints are derived from a protograph-based extrinsic information transfer chart. Our design is more flexible than the previous design, because the integer programming-based design allows reduction of update numbers and performance losses through the constraints function, whereas the previous design requires observation of variable node update behaviour. The authors report the performance of their designs of non-uniform schedules in additive white Gaussian noise (AWGN) and inter-symbol interference (ISI) channels. Particularly, in the ISI channel, the authors’ non-uniform schedules are designed with cooperative decoding between a Bahl-Cocke-Jelinek-Raviv (BCJR) detector and an SC-LDPC decoder.

Sliding Window Decoding for QC-SC-LDPC Codes Under the Constraint of Implementation Complexity

Sliding window decoding has been employed as the decoding scheme of quasi-cyclic spatially-coupled low-density parity-check (QC-SC-LDPC) codes, with the decoding latency and complexity independent of the coupling length but proportional to the window size. The window size was empirically chosen at least three times the constraint length in previous works, which results in the preference of QC-SC-LDPC codes with small constraint length, to reduce the decoding latency and complexity in practical systems. However, the code design freedom and the decoder throughput are limited for these QC-SC-LDPC codes. In this paper, the optimal window size under the constraint of practical decoding complexity is analyzed employing multi-edge-type density evolution (MET-DE). It can be verified from the MET-DE analytical result that the optimal window size could achieve less than twice the constraint length for several edge spreading patterns, which largely loose this restriction in practical code design. Furthermore, it could be observed that besides the constraint length, the specific structure of the edge spreading pattern could affect the optimal window size as well, and thus the MET-DE analytical method for the optimal window size could effectively guide the design of QC-SC-LDPC codes in practical systems. Then an improved sliding window decoding scheme is proposed, including the optimal choice of window size and a parity-check based early termination scheme. Finally, the performance and robustness of the proposed sliding window decoding scheme for QC-SC-LDPC codes are demonstrated via simulation, with the decoding complexity at most twice as that of the conventional LDPC coded schemes.

Optimization of SC-LDPC Codes for Window Decoding With Target Window Sizes

In this paper, we propose an optimization method for protograph-based spatially coupled low-density parity-check (SC-LDPC) codes under window decoding (WD). Previous works on constructing SC-LDPC codes for WD typically focused on optimizing asymptotic performance metrics such as the WD threshold. However, in this paper, it is observed that the WD threshold is not an appropriate metric to sufficiently explain the finite-length behavior of SC-LDPC codes under WD. Thus, we propose a new performance metric, called the window mean parameter, based on a scaling analysis to capture the WD performance more accurately and formulate a code optimization algorithm that optimizes the proposed performance metric. Since the proposed metric depends on the window size, the optimization algorithm can provide a code family of SC-LDPC codes optimized for various target window sizes. Simulation results confirm that the improvement in the proposed metric leads to a finite-length performance improvement, resulting in one to two orders of the frame error rate gain over the conventional SC-LDPC codes for a wide range of window sizes. Furthermore, we investigate structural characteristics of the proposed codes to provide a supplementary explanation for the performance improvement, which also promotes a better understanding of SC-LDPC codes for WD.

Associated Sectors of Magnetic Recording Systems Using Spatially Coupled LDPC Codes

In traditional magnetic recording systems, non-associated sectors are mainly adopted, whereby two consecutive sectors are decoded independently by the low-density parity-check (LDPC) codes. In this paper, we propose a magnetic recording system with associated sectors, constructed using spatially coupled low-density parity-check (SC-LDPC) codes. If the SC-LDPC decoder cannot correct the erroneous bits in the current sector, it can request information stored in previous sectors to improve decoding performance. Moreover, we modify protograph-based extrinsic information transfer (P-EXIT) charts to examine the theoretical performance of SC-LDPC codes applied to both non-associated and associated sectors. Our theoretical results show that the associated sectors achieve significant performance gains compared to the traditional non-associated sectors.

Transmission of High-Frequency Terahertz Band Signal Beyond 300 GHz Over Metallic Hollow Core Fiber

Terahertz band has become a promising research field due to its advantages such as high frequency and large available bandwidth. In this paper, we realized the transmission of the 325 GHz 30 Gbaud terahertz quadrature phase shift keying (QPSK) signal with a bit error ratio (BER) less than the 7% hard-decision forward-error-correction (HD-FEC) threshold of 3.8 &#x00D7; 10^&#x2212;3, and 325 GHz 16 Gbaud 16-ary quadrature amplitude modulation (16QAM) signal with a BER less than the 20% soft-decision forward-error-correction (SD-FEC) threshold of 2.0 &#x00D7; 10^&#x2212;2 over 1 m metallic hollow core fiber. In addition, the performances of the metallic hollow core fiber transmission system under different frequencies from 300 to 380 GHz have also been measured. With the help of the Volterra nonlinearity compensation (VNC) algorithm, the highest net transmission rate can be up to 140.625 Gbit/s for 300 GHz 30 Gbaud 64QAM signal with satisfying the 28% spatially coupled low density parity check (SC-LDPC) threshold of 5.2 &#x00D7; 10^&#x2212;2. By comparing the performances of the metallic hollow core fiber transmission system and wireless transmission system, we also proved the superiority of the hollow core fiber to transmit terahertz signals. The experimental results show that compared with the commonly used free space transmission, the hollow core fiber can be a promising candidate to transmit high-frequency terahertz signals with a high speed and a low loss, and is worthy of further study in the future.

Concatenated Spatially Coupled LDPC Codes With Sliding Window Decoding for Joint Source-Channel Coding

In this paper, a method for joint source-channel coding (JSCC) based on concatenated spatially coupled low-density parity-check (SC-LDPC) codes is investigated. A construction consisting of two SC-LDPC codes is proposed: one for source coding and the other for channel coding, with a joint belief propagation-based decoder. Also, a novel windowed decoding (WD) scheme is presented with significantly reduced latency and complexity requirements. The asymptotic behavior for various graph node degrees is analyzed using a protograph-based Extrinsic Information Transfer (EXIT) chart analysis for both LDPC block codes with block decoding and for SC-LDPC codes with the WD scheme, showing robust performance for concatenated SC-LDPC codes. Simulation results show a notable performance improvement compared to existing state-of-the-art JSCC schemes based on LDPC codes with comparable latency and complexity constraints.

On the Global and Local Decoding of q-Ary SC-LDPC Codes With Moderate Coupling Length

This letter discusses a special type of q-ary spatially coupled low-density parity-check (SC-LDPC) codes with the potential for local decoding (LD). We demonstrate that q-ary SC-LDPCL (suffix “L” stands for locality) code ensembles have better decoding thresholds than the corresponding binary ensembles. We also develop local window decoding (LWD) to improve local decoding performance. The q-ary codes are constructed from low-order finite fields that guarantees relatively low complexity. The provided examples show that q-ary SC-LDPCL codes outperform the corresponding binary codes in both global and local decoding.

On Doped SC-LDPC Codes for Streaming

In streaming applications, doping improves the performance of spatially-coupled low-density parity-check (SC-LDPC) codes by creating reduced-degree check nodes in the coupled chain. We formulate a scaling law to predict the bit and block error rate of periodically-doped semi-infinite SC-LDPC code ensembles streamed over the binary erasure channel under sliding window decoding for a given finite component block length. The scaling law assumes that with some probability doping is equivalent to full termination and triggers two decoding waves; otherwise, decoding performs as if the coupled chain had not been doped at all. We approximate that probability and use the derived scaling laws to predict the error rates of SC-LDPC code ensembles in the presence of doping. The proposed scaling law provides accurate error rate predictions. We further use it to show that in streaming applications periodic doping can yield higher rates than periodic full termination for the same error-correcting performance.

Spatially Coupled LDPC Codes via Partial Superposition and Their Application to HARQ

This paper proposes a new class of spatially coupled codes, which are constructed by sending codewords of a low-density parity-check block code (LDPC-BC) in a block Markov superposition transmission (BMST) manner, resulting in the BMST-LDPC codes. Different from the conventional spatially coupled LDPC (SC-LDPC) codes, the proposed BMST-LDPC codes can have encoder/decoder implemented using the structure of the corresponding LDPC-BCs. The proposed construction is universal , which is applicable to any existing LDPC-BCs including the 5 G New Radio (5G-NR) LDPC-BCs to obtain extra coding gains. The proposed BMST-LDPC codes are also a special class of BMST codes, which have lower error floors even with an encoding memory of one, inheriting a low decoding latency. Moreover, partial superposition , where a fraction of the coded bits are superimposed onto the following codewords, is introduced to alleviate the decoding error propagation. The construction can be optimized, for random long codes, by a protograph-based extrinsic information transfer (EXIT) chart analysis, or by a one-dimensional search for structured block codes. Furthermore, the BMST-LDPC scheme is integrated with the hybrid automatic retransmission request (HARQ) over the block fading channel, improving the throughput performance. Numerical results are presented to validate our analysis and demonstrate the performance advantage of the BMST-LDPC codes over the LDPC-BCs. They also show that the proposed HARQ scheme can yield a throughput improvement of up to 10% over the conventional HARQ scheme.

Nested Array-Based Spatially Coupled LDPC Codes

Linear nested codes, where two or more sub-codes are nested in a global code, have been proposed as candidates for reliable multi-terminal communication. In this article, we consider nested array-based spatially coupled low-density parity-check (SC-LDPC) codes and propose a line-counting based optimization scheme for minimizing the number of dominant absorbing sets in order to improve its performance in the high signal-to-noise ratio regime. Since the parity-check matrices of different nested sub-codes partially overlap, the optimization of one nested sub-code imposes constraints on the optimization of the other sub-codes. To tackle these constraints, a multi-step optimization process is applied first to one of the nested codes, then sequential optimization of the remaining nested codes is carried out based on the constraints imposed by the previously optimized sub-codes. Results show that the order of optimization has a significant impact on the number of dominant absorbing sets in the Tanner graph of the code, resulting in a trade-off between the performance of a nested code structure and its optimization sequence: the code which is optimized without constraints has fewer harmful structures than the code which is optimized with constraints. We also show that for certain code parameters, dominant absorbing sets in the Tanner graphs of all nested codes are completely removed using our proposed optimization strategy.

Construction of Time Invariant Spatially Coupled LDPC Codes Free of Small Trapping Sets

In this paper, we propose a design technique for the construction of variable-regular time-invariant spatially-coupled low-density parity-check (SC-LDPC) codes with small constraint length and low error floor. The proposed technique reduces the error floor by imposing simple constraints on the short cycles in the code's Tanner graph, which in turn, result in the elimination of the most dominant trapping sets of the code. In some cases, we also derive lower bounds on the syndrome former memory for satisfying such constraints. The designed codes are superior to the state-of-the-art in terms of error floor performance and/or decoding complexity and latency.

Girth Analysis and Design of Periodically Time-Varying SC-LDPC Codes

Time-varying spatially coupled low-density parity-check (SC-LDPC) codes with very large period are characterized by significantly better error rate performance and girth properties than their time-invariant counterparts, but the number of parameters they require to be described is usually very large and unpractical. Time-invariant SC-LDPC codes, which can be seen as periodically time-varying codes with unitary period, are represented through a small number of parameters and designed exploiting few degrees of freedom, but their error rate performance and girth properties are sub-optimal. In this paper, we show that the limits of time-invariant SC-LDPC codes can be overcome by transforming them into time-varying SC-LDPC codes with very small period. In particular, we show that periodically time-varying SC-LDPC codes with small period may exhibit significantly better girth properties than the corresponding time-invariant codes by exploiting a larger number of degrees of freedom in the code design, which however scale at most linearly with the product of the code period and the size of the considered base matrix.

Spatially Coupled LDPC Codes With Sub-Block Locality

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 develops a density-evolution analysis for a decoding mode we call semi-global decoding, in which the decoder has access to the requested sub-block plus a prescribed number of sub-blocks around it. SC-LDPCL codes are also studied under a channel model with variability across sub-blocks, for which decoding-performance lower bounds are derived.