Low Decoding Complexity Research Articles

Three Efficient All-Erasure Decoding Methods for Blaum–Roth Codes

Blaum–Roth Codes are binary maximum distance separable (MDS) array codes over the binary quotient ring , where , and p is a prime number. Two existing all-erasure decoding methods for Blaum–Roth codes are the syndrome-based decoding method and the interpolation-based decoding method. In this paper, we propose a modified syndrome-based decoding method and a modified interpolation-based decoding method that have lower decoding complexity than the syndrome-based decoding method and the interpolation-based decoding method, respectively. Moreover, we present a fast decoding method for Blaum–Roth codes based on the LU decomposition of the Vandermonde matrix that has a lower decoding complexity than the two modified decoding methods for most of the parameters.

Multi-User Massive MIMO System with Adaptive Antenna Grouping for Beyond 5G Communication Network

Error-correcting codes with limited errors and higher spectral efficiency are the main concern for wireless communications. In the current situation, research is increasing daily to satisfy the growing demand for users with improved QoS. Adaptive Antenna Grouping (AAG) with a multilevel space–time trellis coding scheme in the Multi-User Massive MIMO system is the better option to provide flexible data transfer speeds, encoding gains, and gain in diversity with improved spectral efficiency and low decoding complexity, including the power optimization by reduced SNR at the same Symbol Error Rate/Frame Error Rate (SER/FER). The prior aim of maintaining spectral efficiency is achieved by using Massive MIMO. This paper presents the AAG according to the channel state information in the Massive MIMO scenario. The impact of the proposed model on standard ITU-R M.2135 scenarios is also demonstrated in this paper.

Hybrid Non-Binary Repeated Polar Codes

Concatenating the state-of-the-art codes at moderate rates with repetition codes has emerged as a practical solution deployed in various standards for ultra-low-power devices such as in Internet-of-Things (IoT) networks. In this paper, we propose a novel concatenation mechanism for such applications which need to operate at very low signal-to-noise ratio (SNR) regime. In the proposed scheme, the outer code is a hybrid polar code constructed in two stages, one with a binary kernel and another also with a binary kernel but applied over a binary extension field. The inner code is a non-binary multiplicative repetition code. This particular structure inherits low-complexity decoding structures of polar codes while enabling concatenation with an inner non-binary multiplicative repetition scheme. The decoding for the proposed scheme is done using cyclic redundancy check (CRC) aided successive cancellation list (SCL) decoder over additive white Gaussian noise (AWGN) and Rayleigh fading channels. Simulation results demonstrate that the proposed hybrid non-binary repeated polar code provides performance gain compared to a polar-repetition scheme with comparable decoding complexity.

Decoding Quadratic Residue Codes Using Deep Neural Networks

In this paper, a low-complexity decoder based on a neural network is proposed to decode binary quadratic residue (QR) codes. The proposed decoder is based on the neural min-sum algorithm and the modified random redundant decoder (mRRD) algorithm. This new method has the same asymptotic time complexity as the min-sum algorithm, which is much lower than the difference on syndromes (DS) algorithm. Simulation results show that the proposed algorithm achieves a gain of more than 0.4 dB when compared to the DS algorithm. Furthermore, a simplified approach based on trapping sets is applied to reduce the complexity of the mRRD. This simplification leads to a slight loss in error performance and a reduction in implementation complexity.

Robust private information retrieval for low encoding/decoding complexity distributed storage

AbstractPrivate information retrieval (PIR) means a user retrieves a file while revealing no information on which file is being retrieved. In classic PIR, the user needs to wait for the responses of all the storage nodes. However, in general, there may be some nodes that are unresponsive, in which case it is formulated as v‐robust PIR in the network coding (NC) structured distributed storage system (DSS), where v denotes the number of unresponsive nodes. We consider PIR with unresponsive nodes under the framework of NC structured DSS. In particular, the storage nodes adopt combination property with zigzag decodable (CP‐ZD) storage code since it has the advantage of extremely low decoding complexity, which is promised by zigzag decoding (ZD) within a binary field. We design a v‐robust PIR scheme with its download communication cost slightly larger than existing studies, but its decoding complexity is significantly reduced when compared with existing studies.

Regional Financial Economic Data Processing Based on Distributed Decoding Technology

To improve the effect of regional financial and economic data processing, this study combines the distributed decoding technology to construct the regional financial data processing system. Furthermore, this study combines the technical characteristics of cooperative communication and unitary space-time codes to design a non-differential distributed space-time code based on cyclic unitary matrix groups. In addition, this study uses the particularity of trigonometric functions to design a non-differential distributed space-time code based on trigonometric functions. Finally, according to the characteristics of the distributed space-time code of trigonometric functions, this study obtains a decoding algorithm with low decoding complexity and uses MATLAB to conduct simulation experiments to verify the financial data processing and economic analysis effect of the system in this study. The experimental analysis results show that the regional financial and economic data processing system based on distributed decoding technology proposed in this study can play an important role in the analysis of financial and economic data.

A Novel Multitask Learning Empowered Codebook Design for Downlink SCMA Networks

Sparse code multiple access (SCMA) is a promising code-domain non-orthogonal multiple access (NOMA) scheme for the enabling of massive machine-type communication. In SCMA, the design of good sparse codebooks and efficient multiuser decoding have attracted tremendous research attention in the past few years. This letter aims to leverage deep learning to jointly design the downlink SCMA encoder and decoder with the aid of autoencoder. We introduce a novel end-to-end learning based SCMA (E2E-SCMA) design framework, under which improved sparse codebooks and low-complexity decoder are obtained. Compared to conventional SCMA schemes, our numerical results show that the proposed E2E-SCMA leads to significant improvements in terms of error rate and computational complexity.

Decoding of Decode and Forward (DF) Relay Protocol using Min-Sum Based Low Density Parity Check (LDPC) System

Decoding high complexity is a major issue to design a decode and forward (DF) relay protocol. Thus, the establishment of low complexity decoding system would beneficial to assist decode and forward relay protocol. This paper reviews existing methods for the min-sum based LDPC decoding system as the low complexity decoding system. Reference lists of chosen articles were further reviewed for associated publications. This paper introduces comprehensive system model representing and describing the methods developed for LDPC based for DF relay protocol. It is consists of a number of components: (1) encoder and modulation at the source node, (2) demodulation, decoding, encoding and modulation at relay node, and (3) demodulation and decoding at the destination node. This paper also proposes a new taxonomy for min-sum based LDPC decoding techniques, highlights some of the most important components such as data used, result performances and profiles the Variable and Check Node (VCN) operation methods that have the potential to be used in DF relay protocol. Min-sum based LDPC decoding methods have the potential to provide an objective measure the best tradeoff between low complexities decoding process and the decoding error performance, and emerge as a cost-effective solution for practical application.

Low-Complexity Chase Decoding of Reed-Solomon Codes Using Channel Evaluation.

A novel time-varying channel adaptive low-complexity chase (LCC) algorithm with low redundancy is proposed, where only the necessary number of test vectors (TVs) are generated and key equations are calculated according to the channel evaluation to reduce the decoding complexity. The algorithm evaluates the error symbol numbers by counting the number of unreliable bits of the received code sequence and dynamically adjusts the decoding parameters, which can reduce a large number of redundant calculations in the decoding process. We provide a simplified multiplicity assignment (MA) scheme and its architecture. Moreover, a multi-functional block that can implement polynomial selection, Chien search and the Forney algorithm (PCF) is provided. On this basis, a high-efficiency LCC decoder with adaptive error-correcting capability is proposed. Compared with the state-of-the-art LCC (TV = 16) decoding, the number of TVs of our decoder was reduced by 50.4% without loss of the frame error rate (FER) performance. The hardware implementation results show that the proposed decoder achieved 81.6% reduced average latency and 150% increased throughput compared to the state-of-the-art LCC decoder.

LDPC decoding with locally informed dynamic scheduling based on the law of large numbers

The informed dynamic scheduling (IDS) strategies, in which the edge message with the maximum message-residual is updated preferentially, achieve remarkable error-correction performance when applied to low-density parity-check (LDPC) codes. However, the IDS strategies incur inferior convergence in iterative decoding owing to the greedy problem, which is called the update-relayed trend here. In order to solve the greediness, two locally informed dynamic scheduling algorithms based on the law of large numbers are proposed. The proposed decoding algorithms use random select of check nodes over a predefined update range (RSPUR) which effectively suppresses the propagation of the update-relayed trend and accordingly restrains the forming of multi-update cycles. Moreover, the decoding algorithm is further improved based on random select of check nodes over an adjustable update range (RSCAR). The update ranges are selected based on the law of large numbers. Therefore, the computational resources can be allocated more equitably by increasing iterations. Simulation results show that both the proposed algorithms achieve excellent performance in terms of throughput and convergence with low decoding complexity over the Additive White Gaussian Noise (AWGN) and the fading channels compared to the previous IDS strategies. Hence, the proposed algorithms behave excellently over the wireless channels.

Low-complexity soft ML detection for generalized spatial modulation

Generalized Spatial Modulation (GSM) is a recent Multiple-Input Multiple-Output (MIMO) scheme, which achieves high spectral and energy efficiencies. Specifically, soft-output detectors have a key role in achieving the highest coding gain when an error-correcting code (ECC) is used. Nowadays, soft-output Maximum Likelihood (ML) detection in MIMO-GSM systems leads to a computational complexity that is unfeasible for real applications; however, it is important to develop low-complexity decoding algorithms that provide a reasonable computational simulation time in order to make a performance benchmark available in MIMO-GSM systems. This paper presents three algorithms that achieve ML performance. In the first algorithm, different strategies are implemented, such as a preprocessing sorting step in order to avoid an exhaustive search. In addition, clipping of the extrinsic log-likelihood ratios (LLRs) can be incorporating to this algorithm to give a lower cost version. The other two proposed algorithms can only be used with clipping and the results show a significant saving in computational cost. Furthermore clipping allows a wide-trade-off between performance and complexity by only adjusting the clipping parameter.

Low-Complexity Voronoi Shaping for the Gaussian Channel

Voronoi constellations (VCs) are finite sets of vectors of a coding lattice enclosed by the translated Voronoi region of a shaping lattice, which is a sublattice of the coding lattice. In conventional VCs, the shaping lattice is a scaled-up version of the coding lattice. In this paper, we design low-complexity VCs with a cubic coding lattice of up to 32 dimensions, in which pseudo-Gray labeling is applied to minimize the bit error rate. The designed VCs have considerable shaping gains of up to 1.03 dB and finer choices of spectral efficiencies in practice compared with conventional VCs. A mutual information estimation method and a log-likelihood approximation method based on importance sampling for very large constellations are proposed and applied to the designed VCs. With error-control coding, the proposed VCs can have higher information rates than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature, with a lower decoding complexity.

Optimum GSSK Transmission in Massive MIMO Systems Using the Box-LASSO Decoder

We propose in this work to employ the Box-LASSO, a variation of the popular LASSO method, as a low-complexity decoder in a massive multiple-input multiple-output (MIMO) wireless communication system. The Box-LASSO is mainly useful for detecting simultaneously structured signals such as signals that are known to be sparse and bounded. One modulation technique that generates essentially sparse and bounded constellation points is the so-called generalized space-shift keying (GSSK) modulation. In this direction, we derive high dimensional sharp characterizations of various performance measures of the Box-LASSO such as the mean square error, probability of support recovery, and the element error rate, under independent and identically distributed (i.i.d.) Gaussian channels that are not perfectly known. In particular, the analytical characterizations can be used to demonstrate performance improvements of the Box-LASSO as compared to the widely used standard LASSO. Then, we can use these measures to optimally tune the involved hyper-parameters of Box-LASSO such as the regularization parameter. In addition, we derive optimum power allocation and training duration schemes in a training-based massive MIMO system. Monte Carlo simulations are used to validate these premises and to show the sharpness of the derived analytical results.

Reduced-Complexity Successive-Cancellation Decoding for Polar Codes on Channels With Insertions and Deletions

In this paper, a simplified successive cancellation (SC) decoding algorithm for polar codes on insertion/deletion error channels is proposed. First, the SC decoding is designed to decode polar codes on insertion/deletion channels and the joint weight distribution is derived to measure the occurrence probability of different scenarios. Some scenarios with small occurrence probability can be pruned to obtain lower decoding complexity with negligible performance loss. Inspired by this, a fixed pruning strategy (FPS) is proposed to reduce the decoding complexity, which can prune as many scenarios as possible with the given performance requirement. By exploiting the periodicity of the joint weight distribution, the upper bound of the block error rate of the pruned SC decoding is derived. Furthermore, according to the convergence of the upper bound, a dynamic self-adjusting pruning strategy is designed to further reduce the decoding complexity and improve the flexibility of the pruning algorithm. Simulation results show that the decoding complexity of the proposed pruning-based decoding algorithms is significantly reduced compared to the state-of-the-art scenario simplified SC decoding algorithm.

Low-Complexity Decoder for Overloaded Uniquely Decodable Synchronous CDMA

We consider the problem of designing a low-complexity decoder for antipodal uniquely decodable (UD) /errorless code sets for overloaded synchronous code-division multiple access (CDMA) systems, where the number of signals <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K_{\mathrm{max}}^{a}$ </tex-math></inline-formula> is the largest known for the given code length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> . In our complexity analysis, we illustrate that compared to maximum-likelihood (ML) decoder, which has an exponential computational complexity for even moderate code lengths, the proposed decoder has a quasi-quadratic computational complexity. Simulation results in terms of bit-error-rate (BER) demonstrate that the performance of the proposed decoder has only a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1-2$ </tex-math></inline-formula> dB degradation in signal-to-noise ratio (SNR) at a BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> when compared to ML. Moreover, we derive the proof of the minimum Manhattan distance of such UD codes and we provide the proofs for the propositions; these proofs constitute the foundation of the formal proof for the maximum number users <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K_{\mathrm{max}}^{a}$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L=8$ </tex-math></inline-formula> .

The Stability of Low-Density Parity-Check Codes and Some of its Consequences

We study the stability of low-density parity-check codes under blockwise or bitwise maximum a posteriori decoding, where transmission takes place over a binary-input memoryless output-symmetric channel. Our study stems from the consideration of constructing universal capacity-achieving codes under low-complexity decoding algorithms, where universality refers to the fact that we are considering a family of channels with equal capacity. Consider, e.g., the right-regular sequence by Shokrollahi and the heavy-tail Poisson sequence by Luby et al. Both sequences are provably capacity-achieving under belief propagation decoding when transmission takes place over the binary erasure channel. In this paper we show that many existing capacity-achieving sequences of low-density parity-check codes are not universal under belief propagation decoding. We reveal that the key to showing this non-universality result is determined by the stability of the underlying codes. More concretely, for an ordered and complete channel family and a sequence of low-density parity-check code ensembles, we determine a stability threshold associated with them, which gives rise to a sufficient condition under which the sequence is not universal under belief propagation decoding. Moreover, we show that the same stability threshold applies to blockwise or bitwise maximum a posteriori decoding as well. We demonstrate how the stability threshold can determine an upper bound on the corresponding blockwise or bitwise maximum a posteriori threshold, revealing the operational significance of the stability threshold.

Design of Bilayer and Multi-Layer LDPC Ensembles From Individual Degree Distributions

A new approach for designing bilayer and multi-layer LDPC codes is proposed and studied in the asymptotic regime. The ensembles are defined through individual uni-variate degree distributions, one for each layer. We present a construction that: 1) enables low-complexity decoding for high-SNR channel instances, 2) provably approaches capacity for low-SNR instances, 3) scales linearly (in terms of design complexity) in the number of layers. For the setup where decoding the second layer is significantly more costly than the first layer, we propose an optimal-cost decoding schedule and study the trade-off between code rate and decoding cost.

Soft-Decision Decoding of Permutation-Based Optical Codes for a Multiple Access System

A multiple access (MA) soft-decision (SD) decoder based on the cost rankings of the channel output matrix, is proposed and applied in an optical wireless communication system in which permutation-based optical codes (POCs) are used. The proposed decoder determines the possible transmitted MA codewords by implementing the Hungarian and Murty's algorithms to solve and rank the costs of all the possible codewords of the received signal. The performance of the proposed Hungarian-Murty (HM) decoder is evaluated in comparison with two decoders based on the Exhaustive Search (ES) technique, in terms of computational complexity and block error rate (BER). It is found that when POCs are employed, the proposed HM decoder achieves better BER performance with a low decoding complexity in comparison to the ES based decoders.

Coded Computing via Binary Linear Codes: Designs and Performance Limits

We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> smaller tasks, encoded using an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,k)$ </tex-math></inline-formula> linear code, and performed over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> distributed nodes. The goal is to reduce the average execution time of the computational job. We provide a connection between the problem of characterizing the average execution time of a coded distributed computing system and the problem of analyzing the error probability of codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> used over erasure channels. Accordingly, we present closed-form expressions for the execution time using binary random linear codes and the best execution time any linear-coded distributed computing system can achieve. It is also shown that there exist <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">good</i> binary linear codes that not only attain (asymptotically) the best performance that any linear code (not necessarily binary) can achieve but also are numerically stable against the inevitable rounding errors in practice. We then develop a low-complexity algorithm for decoding Reed-Muller (RM) codes over erasure channels. Our decoder only involves additions, subtractions, and inversion of relatively small matrices of dimensions at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log n+1$ </tex-math></inline-formula> , and enables coded computation over real-valued data. Extensive numerical analysis of the fundamental results as well as RM- and polar-coded computing schemes demonstrate the excellence of the RM-coded computation in achieving close-to-optimal performance while having a low-complexity decoding and explicit construction. The proposed framework in this paper enables efficient designs of distributed computing systems given the rich literature in the channel coding theory.

A Low-Complexity Decoder With Non-Uniform Quantization for Recursive BMST Codes

Recursive block Markov superposition transmission (rBMST) can be used to construct high-performance spatially coupled codes. When used in power-limited scenarios, low-complexity quantized decoders are required for the rBMST codes. In this letter, we introduce an efficient decoder for rBMST codes by incorporating non-uniform quantization (NUQ) into the threshold attenuated min-sum algorithm (TAMSA). The resulting decoder is referred to as the NUQ-TAMSA. We use the hierarchical dynamic quantization algorithm to design the non-uniform quantizer. To show the effectiveness of the NUQ-TAMSA, we carry out comprehensive comparisons. Our results show that, for low-resolution quantization, performance gains as large as 0.3 dB can be obtained with the NUQ-TAMSA. Numerical results also show that, for a wide range of coding rates, 5-bits non-uniform quantization is sufficient for NUQ-TAMSA to approach the performances of the floating-point based TAMSA.