Soft Decoding Research Articles

Algebraic Soft Decoding of Elliptic Codes

This paper proposes the algebraic soft decoding (ASD) for one-point elliptic codes, where the interpolation problem is solved from the perspective of module basis reduction. In ASD, the interpolation polynomial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {Q}(x, y, z)$ </tex-math></inline-formula> is the minimum candidate of a Gröbner basis. Based on a multiplicity matrix, an interpolation ideal can be defined. With the decoding output list size, an equivalent interpolation module can be led to. By further defining the set of interpolation points, a sequence of modules from the elliptic curve coordinate ring can be obtained. Based on the Lagrange interpolation functions over elliptic function field, a basis of the interpolation module can be constructed. The desired Gröbner basis that contains <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {Q}$ </tex-math></inline-formula> can be determined by reducing the module basis. Re-encoding transform (ReT) is further introduced to reduce the basis reduction complexity. It is also shown that the interpolation can be facilitated by assessing the degree of the Lagrange interpolation polynomials. The decoding complexity is analyzed, which is verified by numerical results. That shows the advantage of this interpolation technique over the conventional Kötter’s interpolation. The ASD performance of elliptic codes is also presented.

NL-CALIC Soft Decoding Using Strict Constrained Wide-Activated Recurrent Residual Network.

In this work, we propose a normalized Tanh activate strategy and a lightweight wide-activate recurrent structure to solve three key challenges of the soft-decoding of near-lossless codes: 1. How to add an effective strict constrained peak absolute error (PAE) boundary to the network; 2. An end-to-end solution that is suitable for different quantization steps (compression ratios). 3. Simple structure that favors the GPU and FPGA implementation. To this end, we propose a Wide-activated Recurrent structure with a normalized Tanh activate strategy for Soft-Decoding (WRSD). Experiments demonstrate the effectiveness of the proposed WRSD technique that WRSD outperforms better than the state-of-the-art soft decoders with less than 5% number of parameters, and every computation node of WRSD requires less than 64KB storage for the parameters which can be easily cached by most of the current consumer-level GPUs. Source code is available at https://github.com/dota-109/WRSD.

Improved DC-Free Run-Length Limited 4B6B Codes for Concatenated Schemes

In this letter, we introduce a class of improved DC-free 4B6B codes in terms of error correction capabilities for a serially concatenated architecture. There are billions of different codebooks that can be derived from the 16 codewords contained in the traditional 4B6B code as per the IEEE 802.15.7 standard for visible light communication (VLC). These codebooks can be classified based on distances properties which determine their error correction performances. The traditional 4B6B code is suitable for hard-decision decoding, however, when a soft decoder is used like in a serially concatenated architecture, that code becomes obsolete. Simulations show that the proposed 4B6B code concatenated with forward error correction (FEC) codes, has better performance compared to state-of-the-art schemes such as the original 4B6B code, the enhanced Miller code, the Manchester code, the 5B10B code and the (0,4) 2/3 RLL code.

Development of method of decoding block codes based on differential evolution procedure

The approach of soft decoding of block codes based on determining the most reliable basis of the generator matrix and applying the differential evolution procedure is proposed. The choice of this search optimization procedure was made as a result of the analysis of the features and limitations of evolutionary optimization procedures. The scheme and the essence of the main stages of the developed method of soft decoding of block codes are presented. At the first stage, a hard decision is formed and the received word syndrome is calculated. After that, the received symbols are ranked by reliability and the generator matrix of the block code is transformed into the corresponding most reliable basis. Next, a differential evolution procedure is applied to search for the most probable transmitted information message and a binary codeword. Decoding is completed by inverse transformation of the found most probable binary codeword by rearranging the corresponding elements. It is shown that the key stage of decoding is the search for the transmitted codeword using the differential evolution procedure, and the formation of the most reliable basis of the generator matrix of block code makes it possible to increase the decoding efficiency. In order to be able to technically implement this decoding method, an appropriate algorithm has been developed and its main steps are given. The results of the work can be used for the implementation of new generation radio communication technologies to improve the reliability of the transmission of service messages. It is also recommended to use the obtained results when solving the problem of decoding other error-correcting code structures that are used in modern telecommunication technologies.

Reliability‐based quantization for soft decoding of LDPC codes in short‐reach optical links

Efficient quantizer design is vital in high-speed optical receivers. In this paper, the authors propose a three-level (1.5-bit) reliability-based quantizer for low-complexity soft decoding of low-density parity-check (LDPC) codes, and show that it outperforms a conventional quantizer employing eight levels (3 bits). Different approaches for obtaining the quantization thresholds are discussed, and it is shown that the achievable channel capacity with the proposed quantizer is within 0.4 dB of the Shannon limit of the unquantized channel. Rate-compatible LDPC codes are also designed tailored to the asymmetric optical channel terminated with the proposed reliability-based quantizer, and it is shown that the resulting quantizer-aware LDPC codes significantly outperform other existing codes of similar rates.

CMDNet: Learning a Probabilistic Relaxation of Discrete Variables for Soft Detection With Low Complexity

Following the great success of Machine Learning (ML), especially Deep Neural Networks (DNNs), in many research domains in 2010s, several ML-based approaches were proposed for detection in large inverse linear problems, e.g., massive MIMO systems. The main motivation behind is that the complexity of Maximum A-Posteriori (MAP) detection grows exponentially with system dimensions. Instead of using DNNs, essentially being a black-box, we take a slightly different approach and introduce a probabilistic Continuous relaxation of disCrete variables to MAP detection. Enabling close approximation and continuous optimization, we derive an iterative detection algorithm: Concrete MAP Detection (CMD). Furthermore, extending CMD by the idea of deep unfolding into CMDNet, we allow for (online) optimization of a small number of parameters to different working points while limiting complexity. In contrast to recent DNN-based approaches, we select the optimization criterion and output of CMDNet based on information theory and are thus able to learn approximate probabilities of the individual optimal detector. This is crucial for soft decoding in today's communication systems. Numerical simulation results in MIMO systems reveal CMDNet to feature a promising accuracy complexity trade-off compared to State of the Art. Notably, we demonstrate CMDNet's soft outputs to be reliable for decoders.

Combining serial and parallel decoding for turbo codes

Reducing the decoding latency of the turbo codes is important to real-time applications. Conventionally, the decoding of the turbo codes (TC) runs in serial fashion, which means only one of the constituent soft decoders runs at a time. Parallel decoding (PD) refers to running the soft decoders in parallel. Although it delivers the output faster (compared to the serial decoding (SD)), it affects the bit- and frame-error rates. This paper proposes a decoding procedure that combines both PD and SD. It bridges the two decoding modes to determine the best combination scheme to achieve the required level of performance at an acceptable decoding latency. Presented results show how this procedure can mitigate the performance degradation at a slight increase in the decoding latency.

Soft decoding of short/medium length codes using ordered statistics for quantum key distribution

Quantum key distribution (QKD) is a cryptographic communication protocol that utilizes quantum mechanical properties for provable absolute security against an eavesdropper. The communication is carried between two terminals using random photon polarization states represented through quantum states. Both these terminals are interconnected through disjoint quantum and classical channels. Information reconciliation using delay controlled joint decoding is performed at the receiving terminal. Its performance is characterized using data and error rates. Achieving low error rates is particularly challenging for schemes based on error correcting codes with short code lengths. This article addresses the decoding process using ordered statistics decoding for information reconciliation of both short and medium length Bose–Chaudhuri–Hocquenghem codes over a QKD link. The link’s quantum channel is modeled as a binary symmetric quantum depolarization channel, whereas the classical channel is configured with additive white Gaussian noise. Our results demonstrate the achievement of low bit error rates, and reduced decoding complexity when compared to other capacity achieving codes of similar length and configuration.

Enhanced stochastic soft decoding algorithm for Reed-Solomon codes

The Chase algorithm based on stochastic computing can significantly reduce the complexity of soft decoding while ensuring the decoding performance, so that the softdecoding algorithm of Reed-Solomon codes is practically applied. However, the time complexity of the stochastic Chase algorithm will increase as the number of test test-vectors increases. In addition, under high-order modulation, the search range of generating test-vectors in the symbol-level stochastic Chase algorithm (SSCA) will increase exponentially by the order of modulation, thus increasing the hardware storage consumption. In order to reduce the time complexity, this paper proposes an early output algorithm, which does not require all test-vectors to achieve successful decoding.Simulation results show that the proposed algorithm can approach the original stochastic decoding performance while reducing the time complexity to nearly $1/\\tau$ ($\\tau$ is the number of test-vectors).In view of the storage issue in the SSCA, this paper reduces the search range by search-radius-determination assisted selection method. Simulation results indicate that the proposed $3\\sigma$-SSCA can reduce the search range of a single symbol of a test-vector from $q$ ($q$ is the order of modulation) to less than ten at the cost of a small performance loss.

A ML-MMSE Receiver for Millimeter Wave User-Equipment Detection: Beamforming, Beamtracking, and Data-Symbols Detection

For a millimeter wave (mmWave) system consisting of a basestation (BS) and a mobile user-equipment (UE), the problem of signal-and-data detection is investigated, and a low complexity ML-MMSE receiver for mmWave beamforming, beamtracking, and data-symbols detection is proposed. Specifically, with a (practical) hybrid beamforming transceiver architecture at both the BS and UE, a multistage angle-of-arrival (AoA) estimation based beamtraining algorithm that provides fast acquisition of the best beam pairs for the BS-UE link is developed. In addition, a novel joint beamtracking and data-symbols detection algorithm, equipped with an adaptive equalizer, is also developed. The algorithm can simultaneously track the best receiving beams, and produce the data estimates that are easy to extract soft bit information for soft decoding; also, it can tackle the dynamic changes of the baseband effective channel. Analytical and simulation results show that the proposed receiver performs well over a broad range of SNR-it can rapidly acquire the most dominant AoAs for beamforming and constantly track the best moving beams due to UE's mobility or device rotation-and in particular, it achieves near-optimal spectral efficiency for a mobile UE with a single RF chain or very few RF chains.

Efficiently computing logical noise in quantum error-correcting codes

Quantum error correction protocols have been developed to offset the high sensitivity to noise inherent in quantum systems. However, much is still unknown about the behavior of a quantum error-correcting code under general noise, including noisy measurements. This lack of knowledge is largely due to the computational cost of simulating quantum systems large enough to perform nontrivial encodings. In this paper, we develop general methods for incorporating noisy measurement operations into simulations of quantum error-correcting codes and show that measurement errors on readout qubits manifest as a renormalization on the effective logical noise. We also derive general methods for reducing the computational complexity of calculating the exact effective logical noise by many orders of magnitude. This reduction is achieved by determining when different recovery operations produce equivalent logical noise. These methods could also be used to better approximate soft decoding schemes for concatenated codes or to reduce the size of a lookup table to speed up the error correction step in implementations of quantum error-correcting codes. We give examples of such reductions for the three-qubit, five-qubit, Steane, concatenated, and toric codes.

Soft Interference Cancellation for Random Coding in Massive Gaussian Multiple-Access.

In 2017, Polyanskiy showed that the trade-off between power and bandwidth efficiency for massive Gaussian random access is governed by two fundamentally different regimes: low power and high power. For both regimes, tight performance bounds were found by Zadik et al., in 2019. This work utilizes recent results on the exact block error probability of Gaussian random codes in additive white Gaussian noise to propose practical methods based on iterative soft decoding to closely approach these bounds. In the low power regime, this work finds that orthogonal random codes can be applied directly. In the high power regime, a more sophisticated effort is needed. This work shows that power-profile optimization by means of linear programming, as pioneered by Caire et al. in 2001, is a promising strategy to apply. The proposed combination of orthogonal random coding and iterative soft decoding even outperforms the existence bounds of Zadik et al. in the low power regime and is very close to the non-existence bounds for message lengths around 100 and above. Finally, the approach of power optimization by linear programming proposed for the high power regime is found to benefit from power imbalances due to fading which makes it even more attractive for typical mobile radio channels.

Combining hard and soft decoders for hypergraph product codes

Hypergraph product codes are a class of constant-rate quantum low-density parity-check (LDPC) codes equipped with a linear-time decoder called small-set-flip (SSF). This decoder displays sub-optimal performance in practice and requires very large error correcting codes to be effective. In this work, we present new hybrid decoders that combine the belief propagation (BP) algorithm with the SSF decoder. We present the results of numerical simulations when codes are subject to independent bit-flip and phase-flip errors. We provide evidence that the threshold of these codes is roughly 7.5% assuming an ideal syndrome extraction, and remains close to 3% in the presence of syndrome noise. This result subsumes and significantly improves upon an earlier work by Grospellier and Krishna (arXiv:1810.03681). The low-complexity high-performance of these heuristic decoders suggests that decoding should not be a substantial difficulty when moving from zero-rate surface codes to constant-rate LDPC codes and gives a further hint that such codes are well-worth investigating in the context of building large universal quantum computers.

A Nonuniform Reference Voltage Optimization Based on Relative-Precision-Loss Ratios in MLC NAND Flash Memory

In Multi-Level-Cell (MLC) NAND flash memory, cell-to-cell interference (CCI) and retention time have become the main noise that degrades the data storage reliability. To mitigate such noise, a relative precision loss (RPL) nonuniform reference voltage sensing strategy is proposed in this paper. First, based on the NAND flash channel model with CCI and retention noise, we simulate the data storage process of MLC NAND flash by Monte Carlo method, and find that the threshold-voltage of each disturbed storage state shows approximately to be Gaussian distributed. Then, by Gaussian approximation, the distribution of threshold voltage can be estimated easily in mathematics with a little loss. Second, we introduce a concept of log-likelihood ratio (LLR)-based RPL ratio to determine the dominating overlap regions, and then propose a new nonuniform reference voltage sensing strategy. This strategy does not only reduce the memory sensing precision (i.e., the number of reference voltages), but also maintains the reliability of the soft information of NAND flash memory channel output for soft decoding. Third, we implement extensive simulations to verify the performance of the new nonuniform sensing strategy. The BER performances of LDPC codes for different sensing strategies are provided to show that the proposed LLR-based RPL-nonuniform sensing strategy can make a good compromise between memory sensing latency and error-correction performance.

Performance Analysis of Physical Layer Network Coding in Massive MIMO Systems With M-QAM Modulations

In this paper, we develop a practical approach for deploying Physical Layer Network Coding (PNC) in multi-user M-Ary Quadrature Amplitude Modulation (M-QAM) Massive Multiple-Input Multiple-Output (MIMO) systems. We formulate a PNC mapping scheme as a function of clusters of estimated summation and difference (SD) of the transmitted symbols from user pairs. Utilizing existing linear detection schemes, such as Zero Forcing (ZF) and Minimum Mean Square Error (MMSE), a cluster of SD symbols are detected using an SD linearly transformed channel matrix. Furthermore, utilizing Maximum a Posteriori (MAP) soft decoding, the SD symbols are mapped to the PNC symbols, leveraging on the PNC symbol that maximizes the likelihood function. For each variant of M-QAM, we derive and simplify a specialization of the generalized PNC mapping function. The error performance results, through simulation, reveal that the proposed PNC scheme achieves twice the spectral efficiency in Massive MIMO, without changing the latter's underlying framework and without any degradation in the bit-error-rate (BER). In fact, our investigation has proved that the BER of the proposed Massive MIMO and PNC is slightly better than that of the conventional Massive MIMO. The feasibility of deploying our proposed PNC scheme in Massive MIMO systems paves way for NC applications to be realized in cellular systems.

Ultra High Fidelity Deep Image Decompression With l∞-Constrained Compression.

We propose a novel asymmetric image compression system of light l∞ -constrained predictive encoding and heavy-duty CNN-based soft decoding. The system achieves superior rate-distortion performances over the best of existing image compression methods, including BPG, WebP, FLIF and recent CNN codecs, in both l2 and l∞ error metrics, for bit rates near or above the threshold of perceptually transparent reconstruction. These remarkable coding gains are made by deep learning for compression artifact removal. A restoration CNN is designed to map a lossy compressed image to its original. Its unique strength is to enforce a tight error bound on a per pixel basis. As such, no small distinctive structures of the original image can be dropped or distorted, even if they are statistical outliers that are otherwise sacrificed by mainstream CNN restoration methods.

Immunity of the Data Transmission Channel with FHSS when Soft Decoding by Means of Spectral Density Estimation of Interference Power

The characteristics of noise immunity of a coherent data transmission channel with Frequency Hopping Spread Spectrum (FHSS), orthogonal signals and Reed-Solomon code with its soft decoding have been obtained using spectral density estimation of interference power, which is a combination of two Gaussian noise interference, they are barrage jamming and one concentrated in the FHSS band part. It is found that when the volume of an orthogonal signals ensemble increases, noise immunity rises due to the accuracy growth of the spectral density estimation of the interference power, which in turn provides an accuracy growth of posterior probability estimation used for soft-decision decoding

Design of Improved BP Decoders and Corresponding LT Code Degree Distribution for AWGN Channels

This paper presents the performance of a hard decision belief propagation (HDBP) decoder used for Luby transform (LT) codes over additive white Gaussian noise channels; subsequently, three improved HDBP decoders are proposed. We first analyze the performance improvement of the sorted ripple and delayed decoding process in a HDBP decoder; subsequently, we propose ripple-sorted belief propagation (RSBP) as well as ripple-sorted and delayed belief propagation (RSDBP) decoders to improve the bit error rate (BER). Based on the analysis of the distribution of error encoded symbols, we propose a ripple-sorted and threshold-based belief propagation (RSTBP) decoder, which deletes low-reliability encoded symbols, to further improve the BER. Degree distribution significantly affects the performance of LT codes. Therefore, we propose a method for designing optimal degree distributions for the proposed decoders. Through simulation results, we demonstrate that the proposed RSBP and RSDBP decoders provide significantly better BER performances than the HDBP decoder. RSDBP and RSTDP combined with the proposed degree distributions outperformed state-of-the-art degree distributions in terms of the number of encoded symbols required to recover an input symbol correctly (NERRIC) and the frame error rate (FER). For a hybrid decoder formulated by combining RSDBP with a soft decision belief propagation decoder, the proposed degree distribution outperforms the other degree distributions in terms of decoding complexity.

Low-Complexity Chase Decoding of Reed-Solomon Codes Using Module

The interpolation based algebraic soft decoding yields a high decoding performance for Reed-Solomon (RS) codes with a polynomial-time complexity. Its computationally expensive interpolation can be facilitated using the module structure. The desired Grobner basis can be achieved by reducing the basis of a module. This paper proposes the low-complexity Chase (LCC) decoding algorithm using this module basis reduction (BR) interpolation technique, namely the LCC-BR algorithm. By identifying $\eta $ unreliable symbols, $2^\eta $ decoding test-vectors will be formulated. The LCC-BR algorithm first constructs a common basis which will be shared by the decoding of all test-vectors. This eliminates the redundant computation in decoding each test-vector, resulting in a lower decoding complexity and latency. This paper further proposes the progressive LCC-BR algorithm that decodes the test-vectors sequentially and terminates once the maximum-likelihood decision decoding outcome is reached. Exploiting the difference between the adjacent test-vectors, this progressive decoding is realized without any additional memory cost. Complexity analysis shows that the LCC-BR algorithm yields a lower complexity and latency, especially for high rate codes, which will be validated by the numerical results.

Iterative Soft Decoding of Reed-Solomon Codes Based on Deep Learning

In this letter, a deep learning based iterative soft decision decoding algorithm for Reed-Solomon codes is proposed. This algorithm takes advantage of deep neural network and Stochastic Shifting based Iterative Decoding (SSID). By assigning weights to every edge in the Tanner graph and changing the architecture of SSID method, the proposed neural network decoder achieves better decoding performance. Compared with BM decoding, HDD-LCC and traditional SSID, simulation results show that for RS (15, 11), this algorithm provides coding gain up to 1.5dB, 1.1dB and 0.5dB, respectively when FER = 10−2.