In this work, we propose a novel decoding algorithm for short block codes based on an edge-weighted graph neural network (EW-GNN). The EW-GNN decoder operates on the Tanner graph with an iterative message-passing structure, which algorithmically aligns with the conventional belief propagation (BP) decoding method. In each iteration, the "weight" on the message passed along each edge is obtained from a fully connected neural network that has the reliability information from nodes/edges as its input. Compared to existing deep-learning-based decoding schemes, the EW-GNN decoder is characterised by its scalability, meaning that 1) the number of trainable parameters is independent of the codeword length, and 2) an EW-GNN decoder trained with shorter/simple codes can be directly used for longer/sophisticated codes of different code rates. Furthermore, simulation results show that the EW-GNN decoder outperforms the BP and deep-learning-based BP methods from the literature in terms of the decoding error rate.
Polar codes, discovered by Arıkan, are the first error-correcting codes with an explicit construction to provably achieve channel capacity, asymptotically. However, their error-correction performance at finite lengths tends to be lower than existing capacity-approaching schemes. Using the successive-cancellation algorithm, polar decoders can be designed for very long codes, with low hardware complexity, leveraging the regular structure of such codes. We present an architecture and an implementation of a scalable hardware decoder based on this algorithm. This design is shown to scale to code le…
Department of Electrical Engineering, Indian Institute of Technology Bombay, India（印度理工学院孟买分校电气工程系）
In this paper, we propose a low latency, robust and scalable neural net based decoder for convolutional and low-density parity-check (LPDC) coding schemes. The proposed decoders are demonstrated to have bit error rate (BER) and block error rate (BLER) performances at par with the state-of-the-art neural net based decoders while achieving more than 8 times higher decoding speed. The enhanced decoding speed is due to the use of convolutional neural network (CNN) as opposed to recurrent neural network (RNN) used in the best known neural net based decoders. This contradicts existing doctrine that …
We propose a scalable video broadcasting scheme over MIMO-OFDM systems. The scalable video source layers are channel encoded and modulated into independent signal streams, which are then transmitted from the allocated antennas in certain time-frequency blocks. Each receiver employs the successive group decoder to decode the signal streams of interest by treating other signal streams as interference. The transmitter performs adaptive coding and modulation, and transmission antenna and subcarrier allocation, based on the rate feedback from the receivers. We also propose a hybrid receiver that sw…
Block Erasure Channel and Block z-Channel with Bounded Decoders and Finite Blocklength
RPTU University Kaiserslautern-Landau（莱茵兰-普法塔尔凯泽斯劳滕-兰道应用技术大学）；Wuhan University（武汉大学）；RWTH Aachen University（亚琛工业大学）；TU Dresden（德累斯顿工业大学）；TU Berlin（柏林工业大学）；Princeton University（普林斯顿大学）；German Research Center for Artificial Intelligence (DFKI)（德国人工智能研究中心）
This paper presents two new space time block codes (STBCs) with quasi orthogonal structure for an open loop multi-input single-output (MISO) systems. These two codes have been designed to transmit from three or four antennas at the transmitter and be given to one antenna at the receiver. First, the proposed codes are introduced and their structures are investigated. This is followed by the demonstration of how the decoder decodes half of transmitted symbols independent of the other half. The last part of this paper discusses the simulation results, makes performance comparison against other po…
Soft-decision decoding is NP-hard problem of great interest to developers of communication system. We present an efficient soft-decision decoding of linear block codes based on compact genetic algorithm (cGA) and compare its performance with various other decoding algorithms including Shakeel algorithms. The proposed algorithm uses the dual code in contrast to Shakeel algorithm that uses the code itself. Hence, this new approach reduces the decoding complexity of high rates codes. The complexity and an optimized version of this new algorithm is also presented and discussed.
This paper reviews the potential channel decoding techniques for ultra-reliable low-latency communications (URLLC). URLLC is renowned for its stringent requirements including ultra-reliability, low end-to-end transmission latency, and packet-size flexibility. These requirements exacerbate the difficulty of the physical-layer design, particularly for the channel coding and decoding schemes. To satisfy the requirements of URLLC, decoders must exhibit superior error-rate performance \black{and} low decoding complexity. \black{Also, it is desired that decoders be universal} to accommodate various …
This letter introduces two fast maximum-likelihood (ML) detection methods for 4*4 quasi-orthogonal space-time block code (QOSTBC). The first algorithm with a relatively simple design exploits structure of quadrature amplitude modulation (QAM) constellations to achieve its goal and the second algorithm, though somewhat more complex, can be applied to any arbitrary constellation. Both decoders utilize a novel decomposition technique for ML metric which divides the metric into independent positive parts and a positive interference part. Search spaces of symbols are substantially reduced by employ…
\begin{abstract} In this paper we consider Time-Varying Block (TVB) codes, which generalize a number of previous synchronization error-correcting codes. We also consider various practical issues related to MAP decoding of these codes. Specifically, we give an expression for the expected distribution of drift between transmitter and receiver due to synchronization errors. We determine an appropriate choice for state space limits based on the drift probability distribution. In turn, we obtain an expression for the decoder complexity under given channel conditions in terms of the state space limi…
Construction of high rate Space Time Block Codes (STBCs) with low decoding complexity has been studied widely using techniques such as sphere decoding and non Maximum-Likelihood (ML) decoders such as the QR decomposition decoder with M paths (QRDM decoder). Recently Ren et al., presented a new class of STBCs known as the block orthogonal STBCs (BOSTBCs), which could be exploited by the QRDM decoders to achieve significant decoding complexity reduction without performance loss. The block orthogonal property of the codes constructed was however only shown via simulations. In this paper, we give …
Full-rate space time codes (STC) with rate = number of transmit antennas have high multiplexing gain, but high decoding complexity even when decoded using reduced-complexity decoders such as sphere or QRDM decoders. In this paper, we introduce a new code property of STC called block-orthogonal property, which can be exploited by QR-decomposition-based decoders to achieve significant decoding complexity reduction without performance loss. We show that such complexity reduction principle can benefit the existing algebraic codes such as Perfect and DjABBA codes due to their inherent (but previous…
Facebook AI Research（脸书人工智能研究院）；Tel Aviv University（特拉维夫大学）
Neural decoders were shown to outperform classical message passing techniques for short BCH codes. In this work, we extend these results to much larger families of algebraic block codes, by performing message passing with graph neural networks. The parameters of the sub-network at each variable-node in the Tanner graph are obtained from a hypernetwork that receives the absolute values of the current message as input. To add stability, we employ a simplified version of the arctanh activation that is based on a high order Taylor approximation of this activation function. Our results show that fo…
We propose a new type of short to moderate block-length, linear error-correcting codes, called moderate-density parity-check (MDPC) codes. The number of ones of the parity-check matrix of the codes presented is typically higher than the number of ones of the parity-check matrix of low-density parity-check (LDPC) codes. But, still lower than those of the parity-check matrix of classical block codes. The proposed MDPC codes are cyclic and are designed by constructing idempotents using cyclotomic cosets. The construction is simple and allows finding short block-length, high-rate codes with good m…
With the use of belief propagation (BP) decoding algorithm, low-density parity-check (LDPC) codes can achieve near-Shannon limit performance. In order to evaluate the error performance of LDPC codes, simulators running on CPUs are commonly used. However, the time taken to evaluate LDPC codes with very good error performance is excessive. In this paper, efficient LDPC block-code decoders/simulators which run on graphics processing units (GPUs) are proposed. We also implement the decoder for the LDPC convolutional code (LDPCCC). The LDPCCC is derived from a pre-designed quasi-cyclic LDPC block c…
The training complexity of deep learning-based channel decoders scales exponentially with the codebook size and therefore with the number of information bits. Thus, neural network decoding (NND) is currently only feasible for very short block lengths. In this work, we show that the conventional iterative decoding algorithm for polar codes can be enhanced when sub-blocks of the decoder are replaced by neural network (NN) based components. Thus, we partition the encoding graph into smaller sub-blocks and train them individually, closely approaching maximum a posteriori (MAP) performance per sub-…
Low-rate and short-packet transmissions are important for ultra-reliable low-latency communications (URLLC). In this paper, we put forth a new family of sparse superposition codes for URLLC, called block orthogonal sparse superposition (BOSS) codes. We first present a code construction method for the efficient encoding of BOSS codes. The key idea is to construct codewords by the superposition of the orthogonal columns of a dictionary matrix with a sequential bit mapping strategy. We also propose an approximate maximum a posteriori probability (MAP) decoder with two stages. The approximate MAP …
The generalized Poor-Verdu error lower bound established in [1] for multihypothesis testing is studied in the classical channel coding context. It is proved that for any sequence of block codes sent over the memoryless binary symmetric channel (BSC), the minimum probability of error (under maximum likelihood decoding) has a relative deviation from the generalized bound that grows at most linearly in blocklength. This result directly implies that for arbitrary codes used over the BSC, decoder ties can only affect the subexponential behavior of the minimum probability of error.
In this paper, we design the joint decoding (JD) of non-orthogonal multiple access (NOMA) systems employing short block length codes. We first proposed a low-complexity soft-output ordered-statistics decoding (LC-SOSD) based on a decoding stopping condition, derived from approximations of the a-posterior probabilities of codeword estimates. Simulation results show that LC-SOSD has the similar mutual information transform property to the original SOSD with a significantly reduced complexity. Then, based on the analysis, an efficient JD receiver which combines the parallel interference cancellat…
Sparse regression codes (SPARC) connect the sparse signal recovery framework of compressive sensing with error control coding techniques. SPARC encoding produces codewords which are \emph{sparse} linear combinations of columns of a dictionary matrix. SPARC decoding is accomplished using sparse signal recovery algorithms. We construct dictionary matrices using Gold codes and mutually unbiased bases and develop suitable generalizations of SPARC (GSPARC). We develop a greedy decoder, referred as match and decode (MAD) algorithm and provide its analytical noiseless recovery guarantees. We propose …
In this letter, we propose parallel SC (Successive Cancellation) decoder and parallel SC-List decoder for polar codes. The parallel decoder is composed of M=2^m(m>=1) component decoders working in parallel and each component decoder decodes a Polar code of a block size of 1/M of the original Polar code. Therefore the parallel decoder has M times faster decoding speed. Our simulation results show that the parallel decoder has almost the same error-rate performance as the conventional non-parallel decoder.
We study spatially coupled LDPC codes that allow access to sub-blocks much smaller than the full code block. Sub-block access is realized by a semi-global decoder that decodes a chosen target sub-block by only accessing the target, plus a prescribed number of helper sub-blocks adjacent in the code chain. This paper develops a theoretical methodology for analyzing the semi-global decoding performance of spatially coupled LDPC codes constructed from protographs. The main result shows that semi-global decoding thresholds can be derived from certain thresholds we define for the single-sub-block gr…
Iterative decoding techniques have gain popularity due to their performance and their application in most communications systems. In this paper, we present a new application of our iterative decoder on the GPCB (Generalized Parallel Concatenated Block codes) which uses cyclic permutations. We introduce a new variant of the component decoder. After extensive simulation; the obtained result is very promising compared with several existing methods. We evaluate the effects of various parameters component codes, interleaver size, block size, and the number of iterations. Three interesting results a…
In this paper, we present the block Markov superposition transmission of BCH (BMST-BCH) codes, which can be constructed to obtain a very low error floor. To reduce the implementation complexity, we design a low complexity iterative sliding-window decoding algorithm, in which only binary and/or erasure messages are processed and exchanged between processing units. The error floor can be predicted by a genie-aided lower bound, while the waterfall performance can be analyzed by the density evolution method. To evaluate the error floor of the constructed BMST-BCH codes at a very low bit error rate…
This paper presents a new VLSI friendly framework for scalable video coding based on Compressed Sensing (CS). It achieves scalability through 3-Dimensional Discrete Wavelet Transform (3-D DWT) and better compression ratio by exploiting the inherent sparsity of the high-frequency wavelet sub-bands through CS. By using 3-D DWT and a proposed adaptive measurement scheme called AMS at the encoder, one can succeed in improving the compression ratio and reducing the complexity of the decoder. The proposed video codec uses only 7% of the total number of multipliers needed in a conventional CS-based v…
Guessing Random Additive Noise Decoding (GRAND) is a family of universal decoding algorithms suitable for decoding any moderate redundancy code of any length. We establish that, through the use of list decoding, soft-input variants of GRAND can replace the Chase algorithm as the component decoder in the turbo decoding of product codes. In addition to being able to decode arbitrary product codes, rather than just those with dedicated hard-input component code decoders, results show that ORBGRAND achieves a coding gain of up to 0.7dB over the Chase algorithm with same list size.
The problem of designing good Space-Time Block Codes (STBCs) with low maximum-likelihood (ML) decoding complexity has gathered much attention in the literature. All the known low ML decoding complexity techniques utilize the same approach of exploiting either the multigroup decodable or the fast-decodable (conditionally multigroup decodable) structure of a code. We refer to this well known technique of decoding STBCs as Conditional ML (CML) decoding. In this paper we introduce a new framework to construct ML decoders for STBCs based on the Generalized Distributive Law (GDL) and the Factor-grap…
Polar codes with memory (PCM) are proposed in this paper: a pair of consecutive code blocks containing a controlled number of mutual information bits. The shared mutual information bits of the succeeded block can help the failed block to recover. The underlying polar codes can employ any decoding scheme such as the successive cancellation (SC) decoding (PCM-SC), the belief propagation (BP) decoding (PCM-BP), and the successive cancellation list (SCL) decoding (PCM-SCL). The analysis shows that the packet error rate (PER) of PCM decreases to the order of PER squared while maintaining the same c…
In this paper, we develop an efficient decoder via the proximal alternating direction method of multipliers (proximal-ADMM) technique for nonbinary linear block codes in the Galois field. Its main contents are as follows: first, exploiting the decomposition technique based on the three-variables check equation, we formulate the maximum likelihood (ML) decoding problem approximately to a non-convex quadratic program; second, an efficient algorithm based on the proximal-ADMM technique is proposed to solve the formulated QP problem. Exploiting the QP problem's inherent structures, its variables c…
In this work, binary block-to-block distribution matching is considered. m independent and uniformly distributed bits are mapped to n output bits resembling a target product distribution. A rate R is called achieved by a sequence of encoder-decoder pairs, if for m,n to infinity, (1) m/n approaches R, (2) the informational divergence per bit of the output distribution and the target distribution goes to zero, and (3) the probability of erroneous decoding goes to zero. It is shown that the maximum achievable rate is equal to the entropy of the target distribution. A practical encoder-decoder pai…
Conventional turbo codes (CTCs) usually employ a block-oriented interleaving so that each block is separately encoded and decoded. As interleaving and de-interleaving are performed within a block, the message-passing process associated with an iterative decoder is limited to proceed within the corresponding range. This paper presents a new turbo coding scheme that uses a special interleaver structure and a multiple-round early termination test involving both sign check and a CRC code. The new interleaver structure is naturally suited for high speed parallel processing and the resulting coding …
Linear space-time block codes (STBCs) of unitary rate and full diversity, systematically constructed over arbitrary constellations for any number of transmit antennas are introduced. The codes are obtained by generalizing the existing ABBA STBCs, a.k.a quasi-orthogonal STBCs (QO-STBCs). Furthermore, a fully orthogonal (symbol-by-symbol) decoder for the new generalized ABBA (GABBA) codes is provided. This remarkably low-complexity decoder relies on partition orthogonality properties of the code structure to decompose the received signal vector into lower-dimension tuples, each dependent only on…
This paper propose a decoder architecture for low-density parity-check convolutional code (LDPCCC). Specifically, the LDPCCC is derived from a quasi-cyclic (QC) LDPC block code. By making use of the quasi-cyclic structure, the proposed LDPCCC decoder adopts a dynamic message storage in the memory and uses a simple address controller. The decoder efficiently combines the memories in the pipelining processors into a large memory block so as to take advantage of the data-width of the embedded memory in a modern field-programmable gate array (FPGA). A rate-5/6 QC-LDPCCC has been implemented on an …
In this letter, we propose an adaptive SC (Successive Cancellation)-List decoder for polar codes with CRC. This adaptive SC-List decoder iteratively increases the list size until the decoder outputs contain at least one survival path which can pass CRC. Simulation shows that the adaptive SC-List decoder provides significant complexity reduction. We also demonstrate that polar code (2048, 1024) with 24-bit CRC decoded by our proposed adaptive SC-List decoder with very large list size can achieve a frame error rate FER=0.001 at Eb/No=1.1dB, which is about 0.2dB from the information theoretic lim…
Just as the Hamming weight spectrum of a linear block code sheds light on the performance of a maximum likelihood decoder, the pseudo-weight spectrum provides insight into the performance of a linear programming decoder. Using properties of polyhedral cones, we find the pseudo-weight spectrum of some short codes. We also present two general lower bounds on the minimum pseudo-weight. The first bound is based on the column weight of the parity-check matrix. The second bound is computed by solving an optimization problem. In some cases, this bound is more tractable to compute than previously know…
Thompson's model of VLSI computation relates the energy of a computation to the product of the circuit area and the number of clock cycles needed to carry out the computation. It is shown that for any family of circuits implemented according to this model, using any algorithm that performs decoding of a codeword passed through a binary erasure channel, as the block length approaches infinity either (a) the probability of block error is asymptotically lower bounded by 1/2 or (b) the energy of the computation scales at least as Omega(n(log n)^(1/2)), and so the energy of successful decoding, per…
Polar codes have been gaining a lot of interest due to it being the first coding scheme to provably achieve the symmetric capacity of a binary memoryless channel with an explicit construction. However, the main drawback of polar codes is the low throughput of its successive cancellation (SC) decoding. Simplified SC decoding algorithms of polar codes can be used to reduce the latency of the polar decode by faster processing of specific sub-codes in the polar code. By combining simplified SC with a list decoding technique, such as SC list (SCL) decoding, polar codes can cater to the two conflict…
Error correction techniques traditionally focus on the co-design of restricted code-structures in tandem with code-specific decoders that are computationally efficient when decoding long codes in hardware. Modern applications are, however, driving demand for ultra-reliable low-latency communications (URLLC), rekindling interest in the performance of shorter, higher-rate error correcting codes, and raising the possibility of revisiting universal, code-agnostic decoders. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reli…
Under successive cancellation (SC) decoding, polar codes are inferior to other codes of similar blocklength in terms of frame error rate. While more sophisticated decoding algorithms such as list- or stack-decoding partially mitigate this performance loss, they suffer from an increase in complexity. In this paper, we describe a new flavor of the SC decoder, called the SC flip decoder. Our algorithm preserves the low memory requirements of the basic SC decoder and adjusts the required decoding effort to the signal quality. In the waterfall region, its average computational complexity is almost …
In this paper, we study the tradeoffs between complexity and reliability for decoding large linear block codes. We show that using artificial neural networks to predict the required order of an ordered statistics based decoder helps in reducing the average complexity and hence the latency of the decoder. We numerically validate the approach through Monte Carlo simulations.
Polar codes have received growing attention in the past decade and have been selected as the coding scheme for the control channel in the fifth generation (5G) wireless communication systems. However, the conventional polar codes have only been constructed by binary (2x2) kernel which poses block length limitation to powers of 2. To attain more flexible block lengths, multi-kernel polar codes are proposed. In this paper, a combinational architecture for multi-kernel polar codes with high throughput is proposed based on successive cancellation decoding algorithm. The proposed scheme can decode …
This brief proposes a hardware implementation architecture for Fano decoding of polarization-adjusted convolutional (PAC) codes. This architecture uses a novel branch metric unit specific to PAC codes. The proposed decoder is tested on FPGA, and its performance is evaluated on ASIC using TSMC 28 nm 0.72 V library. The decoder can be clocked at 500 MHz and reach an average information throughput of 38 Mb/s at 3.5 dB signal-to-noise ratio for a block length of 128 and a code rate of 1/2.
Designing a practical, low complexity, close to optimal, channel decoder for powerful algebraic codes with short to moderate block length is an open research problem. Recently it has been shown that a feed-forward neural network architecture can improve on standard belief propagation decoding, despite the large example space. In this paper we introduce a recurrent neural network architecture for decoding linear block codes. Our method shows comparable bit error rate results compared to the feed-forward neural network with significantly less parameters. We also demonstrate improved performance …
