Soft-decision Decoding Research Articles

1.06 Tbit/s space division multiplexing self-homodyne coherent transmission with high spectral efficiency probabilistic shaping multidimensional modulation

We propose a probabilistic shaping multidimensional (MD) modulation format for self-homodyne coherent transmission systems. The redundancy of the MD signal is separated and transmitted together with the optical pilot tone (PT). The 4D/8D/12D modulation is constructed at the transmitter based on the amplitude translation (AT)-set partitioning (SP) principle and the probabilistic amplitude shaping (PAS) structure. Moreover, the hard-decision (HD) and soft-decision (SD) decoders for MD signals are introduced. We implement 20 GBaud 4D/8D/12D probabilistic shaping (PS) 16384-level quadrature amplitude modulation (QAM) transmission over 22.5 km uncoupled multicore fiber to analyze the performance. The experimental results show that it is possible for the MD signal to provide spectral efficiency (SE) gain up to 3.31 bit/s/Hz/4D-sym. The maximum SE transmitted is up to 57.29 bit/s/Hz, i.e., 17.5 bit/s/Hz/4D-sym, with a bit rate of 1.06 Tbit/s.

Enhancing Erasure Resilience in Reed-Solomon Codes: Theory and Applications in Data Storage Systems

Reed-Solomon (RS) codes are widely utilized in data storage and communication systems due to their robust error detection and correction capabilities. However, traditional RS codes encounter challenges in addressing the increasing data corruption rates demanded by modern storage systems. This paper aims to overcome these limitations by modifying and extending the structure of traditional RS codes, particularly enhancing their recovery capabilities in the face of significant data loss. We begin by reviewing the theoretical foundations of RS codes and existing extension methods and discussing emerging technologies integrated into RS codes. We then present detailed methodologies for RS code encoding, erasure recovery mechanisms, soft decision decoding, and interleaving coding techniques, followed by a series of experiments designed to test the proposed methods. The experimental results indicate that soft-decision decoding outperforms traditional hard-decision decoding under high signal-to-noise ratio conditions, albeit with increased computational complexity as the list of candidate codewords grows. Interleaved Reed-Solomon (IRS) coding offers improved performance under low signal-to-noise ratio conditions but may introduce additional system complexity due to the interleaving and deinterleaving processes. We conclude with a summary of the research findings, a discussion of the study’s limitations, and suggestions for future research directions.

Overcoming the High Error Rate of Composite DNA Letters-Based Digital Storage through Soft-Decision Decoding.

Composite DNA letters, by merging all four DNA nucleotides in specified ratios, offer a pathway to substantially increase the logical density of DNA digital storage (DDS) systems. However, these letters are susceptible to nucleotide errors and sampling bias, leading to a high letter error rate, which complicates precise data retrieval and augments reading expenses. To address this, Derrick-cp is introduced as an innovative soft-decision decoding algorithm tailored for DDS utilizing composite letters. Derrick-cp capitalizes on the distinctive error sensitivities among letters to accurately predict and rectify letter errors, thus enhancing the error-correcting performance of Reed-Solomon codes beyond traditional hard-decision decoding limits. Through comparative analyses in the existing dataset and simulated experiments, Derrick-cp's superiority is validated, notably halving the sequencing depth requirement and slashing costs by up to 22% against conventional hard-decision strategies. This advancement signals Derrick-cp's significant role in elevating both the precision and cost-efficiency of composite letter-based DDS.

Design and simulation of soft decision decoding based on chaotic M-ary spread spectrum system

In this paper, chaos sequence is used to replace the traditional spread spectrum sequence. From the perspective of the communication partner, a new method of generating the initial value of chaotic sequence is proposed, which is more suitable for the multi-base spread spectrum system and easy to extract. In order to further improve the error performance of the system and ensure the integrity of the system, an improved suboptimal soft information extraction method based on log-likelihood ratio (LLR) is proposed in the channel coding module, which can further improve the error performance of the system. On this basis, we choose the synchronization mode of multiple transmission of one reference signal to solve the problem that chaotic sequence is difficult to obtain synchronization because of its aperiodic characteristics. Finally, the system is simulated in the channel with low SNR. The simulation results show that chaotic sequence as spread spectrum code can make the signal transmit accurately in the worse channel environment. The performance gain of applying soft information decoding decision in the multibase spread spectrum system with chaotic sequence as spread spectrum code can reach 5.2 dB compared with the traditional direct spread spectrum system.

Parity Check Codes for Second Order Diversity

Short block length “block” codes are typically not used in wireless standards as soft decision decoding is computationally intensive and hard decision decoding results in performance loss. However, short block lengths are of interest to massive machine-type communications (mMTC) and ultra-reliable low-latency communications (URLLC). In this paper, we propose a diversity preserving enhanced hard-decision decoding scheme for parity check codes (PCC) over Rayleigh fading channels. The proposed flip decoding scheme has linear complexity in the block length. Theoretical analysis and simulation results verify the correctness of the proposed detection scheme.

Improving error-correcting capability in DNA digital storage via soft-decision decoding.

Error-correcting codes (ECCs) employed in the state-of-the-art DNA digital storage (DDS) systems suffer from a trade-off between error-correcting capability and the proportion of redundancy. To address this issue, in this study, we introduce soft-decision decoding approach into DDS by proposing a DNA-specific error prediction model and a series of novel strategies. We demonstrate the effectiveness of our approach through a proof-of-concept DDS system based on Reed-Solomon (RS) code, named as Derrick. Derrick shows significant improvement in error-correcting capability without involving additional redundancy in both in vitro and in silico experiments, using various sequencing technologies such as Illumina, PacBio and Oxford Nanopore Technology (ONT). Notably, in vitro experiments using ONT sequencing at a depth of 7× reveal that Derrick, compared with the traditional hard-decision decoding strategy, doubles the error-correcting capability of RS code, decreases the proportion of matrices with decoding-failure by 229-fold, and amplifies the potential maximum storage volume by impressive 32388-fold. Also, Derrick surpasses 'state-of-the-art' DDS systems by comprehensively considering the information density and the minimum sequencing depth required for complete information recovery. Crucially, the soft-decision decoding strategy and key steps of Derrick are generalizable to other ECCs' decoding algorithms.

Concatenated Forward Error Correction With KP4 and Single Parity Check Codes

Concatenated forward error correction is studied using an outer KP4 Reed-Solomon code with hard-decision decoding and inner single parity check (SPC) codes with Chase/Wagner soft-decision decoding. Analytical expressions are derived for the end-to-end frame and bit error rates for transmission over additive white Gaussian noise channels with binary phase-shift keying (BPSK) and quaternary amplitude shift keying (4-ASK), as well as with symbol interleavers and quantized channel outputs. The BPSK error rates are compared to those of two other inner codes: a two-dimensional product code with SPC component codes and an extended Hamming code. Simulation results for unit-memory inter-symbol interference channels and 4-ASK are also presented. The results show that the coding schemes achieve similar error rates, but SPC codes have the lowest complexity and permit flexible rate adaptation.

A Hybrid Scheme Combining Duplications and LDPC Decoding to Reduce NAND Flash

Abstract With the development of NAND flash, the storage density has become larger and larger, but the reliability has become lower and lower. To address this problem, stronger error correction codes, low-density parity-check (LDPC) codes, are widely used in flash memory. However, the direct use of LDPC codes for error correction has a great impact on the read and write performance of SSDs. Given this, this paper proposes a new mentality that combines replica and LDPC decoding methods to improve the read performance of solid state drives (SSDs). Considering the characteristics of the quantization level of LDPC soft-decision decoding, this paper proposes the IDSD (LDPC soft-decision decoding and duplication method) and IDHSD (a method combining duplication and LDPC hybrid decoding)schemes. These schemes are mainly for hot data. When reading the data, the IDSD scheme needs to first judge whether it is soft-decision decoding. Then, according to the number of quantization levels recorded, it needs to select whether to use duplication or soft-decision decoding to correct the error data. The IDHSD scheme does not need to judge the soft-decision decoding but directly uses the duplication after the hard-decision decoding fails. Both schemes reduce the average response time to a certain extent and optimize the read performance of the SSD. The experimental results show that compared with the Native Scheme, the average response delay time of the IDSD scheme decreases by 5–30% and the IDHSD scheme slashes by 5–38%. Compared with the Traditional Refresh Scheme, the average response delay time of the IDSD scheme cuts down by 1–30% and the IDHSD scheme lowers by 1–29%.

High-Performance Soft Decision Decoding for Compound Channel Using RS-SPC Concatenated Codes

A novel single parity check-assisted ordered statistic decoding (SPC-OSD) algorithm for the compound channel is proposed for Reed-Solomon (RS) codes. The algorithm can effectively use the channel soft information to accurately determine the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N - K</i> unreliable symbol positions to be erased by using single parity checking and threshold comparison. Besides that, the second-burst error symbol is also proposed as a supplement to unreliable symbols. Based on these predetermined errors, the algorithm can be dynamically adjusted according to the channel environment to achieve the best decoding performance. Compared with burst-error correcting algorithm-based ordered-statistics decoding (BC-OSD) and burst error correcting hard-decision decoding-based low-complexity chase algorithm (BCHDD-LCC), simulation results show that the proposed algorithm can give a coding gain reaching up to 2.0dB and 2.1dB respectively, when symbol error rate (SER) is 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> .

A Simple Neural-Network-Based Decoder for Short Binary Linear Block Codes

The conventional soft decision decoding (SDD) methods require various hard decision decoders (HDDs) based on different codes or re-manipulate the generator matrix by the complicated Gaussian elimination technique according to the bit reliability. This paper presents a general multi-class neural network (NN)-based decoder for the short linear block codes, where no HDD and Gaussian elimination are required once the NN is constructed. This network architecture performs multi-classification to select the messages with high occurrence probabilities and chooses the best codeword on a maximum likelihood basis. Simulation results show that the developed approach outperforms the existing deep neural network (DNN)-based decoders in terms of decoding time and bit error rate (BER). The error-correcting performance is also superior to the conventional Chase-II algorithm and is close to the ordered statistics decoding (OSD) in most cases. For Bose–Chaudhuri–Hocquenghem (BCH) codes, the SNR is improved by 1dB to 4dB as the BER is 10−4. For the (23, 12) quadratic residue (QR) code, the SNR is improved by 2dB when the BER is 10−3. The developed NN-based decoder is quite general and applicable to various short linear block codes with good BER performance.

Novel Real-Time Decision-Based Carrier Tracking for Software-Defined Radios Using M-ARY QAM Modulation

In this paper, a scalable and real-time decision-based carrier tracking for software defined radio (SDR) for M-ary QAM modulation has been proposed and tested in real-time using the universal software radio peripheral (USRP)-2922. The proposed system provides real-time decision-based carrier tracking for improved efficiency and performance. The real-time transfer of data has been done using the USRP-2922 hardware using the M-ary QAM modulation scheme for data transmission. The novelty of the proposed system is that one can transfer random data (binary bits), text or an image which can be encoded using the desired forward error correction codes (FEC) namely the convolution coding/Viterbi soft decision decoding and the turbo coding/decoding. The proposed novel SDR transceiver system with real-time decision-based carrier tracking shows an improved performance for different parameters such as the bit error rate (BER) and signal to noise ratio (Eb/No).

Hybrid HMM Decoder for Convolutional Codes by Joint Trellis-Like Structure and Channel Prior

The anti-interference capability of wireless links is a physical layer problem for edge computing. Although convolutional codes have inherent error correction potential due to the redundancy introduced in the data, the performance of the convolutional code is drastically degraded due to multipath effects on the channel. In this paper, we propose the use of a Hidden Markov Model (HMM) for the reconstruction of convolutional codes and decoding by the Viterbi algorithm. Furthermore, to implement soft-decision decoding, the observation of HMM is replaced by Gaussian mixture models (GMM). Our method provides superior error correction potential than the standard method because the model parameters contain channel state information (CSI). We evaluated the performance of the method compared to standard Viterbi decoding by numerical simulation. In the multipath channel, the hybrid HMM decoder can achieve a performance gain of 4.7 dB and 2 dB when using hard-decision and soft-decision decoding, respectively. The HMM decoder also achieves significant performance gains for the RSC code, suggesting that the method could be extended to turbo codes.

A Low-Complexity RS-SPC Product Coding Scheme for Optical Systems

This paper presents a product coding scheme based on binary images of Reed-Solomon (RS) codes and single-parity-check (SPC) codes for high-speed communications. Utilizing the special selection of components, we propose hybrid soft- and hard-decision iterative decoding (ID) algorithms for these codes, during which many RS component decoders can be early terminated. The proposed hybrid ID only exchanges ternary messages between component decoders and early stops the RS component decoders satisfying a proposed stopping criterion, thus low hardware and computational complexities are guaranteed. The hybrid ID of RS-SPC product codes not only performs comparable to the soft-decision decoding of low-density parity-check (LDPC) codes and block turbo codes (BTCs), but also takes much lower computational complexity and smaller decoder data flow. We also propose semi-analytical methods to estimate the waterfall and error-floor performances of the proposed scheme, which yield relatively accurate estimated results. The simulation results and error-floor estimations indicate that some good RS-SPC product codes have very low error floors that satisfy the requirements of optical systems.

Improved Soft-Aided Decoding of Product Codes With Dynamic Reliability Scores

Products codes (PCs) are conventionally decoded with efficient iterative bounded-distance decoding (iBDD) based on hard-decision channel outputs which entails a performance loss compared to a soft-decision decoder. Recently, several hybrid algorithms have been proposed aimed to improve the performance of iBDD decoders via the aid of a certain amount of soft information while keeping the decoding complexity similarly low as in iBDD. We propose a novel hybrid low-complexity decoder for PCs based on error-and-erasure (EaE) decoding and dynamic reliability scores (DRSs). This decoder is based on a novel EaE component code decoder, which is able to decode beyond the designed distance of the component code but suffers from an increased miscorrection probability. The DRSs, reflecting the reliability of a codeword bit, are used to detect and avoid miscorrections. Simulation results show that this policy can reduce the miscorrection rate significantly and improves the decoding performance. The decoder requires only ternary message passing and a slight increase of computational complexity compared to iBDD, which makes it suitable for high-speed communication systems. Coding gains of up to 1.2 dB compared to the conventional iBDD decoder are observed.

On Hard and Soft Decision Decoding of BCH Codes

The binary primitive BCH codes are cyclic and are constructed by choosing a subset of the cyclotomic cosets. Which subset is chosen determines the dimension, the minimum distance and the weight distribution of the BCH code. We construct possible BCH codes and determine their coderate, true minimum distance and the non-equivalent codes. A particular choice of cyclotomic cosets gives BCH codes which are, extended by one bit, equivalent to Reed-Muller codes, which is a known result from the sixties. We show that BCH codes have possibly better parameters than Reed-Muller codes, which are related in recent publications to polar codes. We study the decoding performance of these different BCH codes using information set decoding based on minimal weight codewords of the dual code. We show that information set decoding is possible even in case of a channel without reliability information since the decoding algorithm inherently calculates reliability information. Different BCH codes of the same rate are compared and different decoding performances and complexity are observed. Some examples of hard decision decoding of BCH codes have the same decoding performance as maximum likelihood decoding. All presented decoding methods can possibly be extended to include reliability information of a Gaussian channel for soft decision decoding. We show simulation results for soft decision list information set decoding and compare the performance to other methods.

Efficient and Low-Complexity Rate and Dimming Control of VLC for Industrial IoT Applications

Visible light communication (VLC) has emerged as a promising complementary technology to millimeter-wave communications for short-range communications to meet the increasing industrial Internet of Things (IIoT) capacity demands. Efficient illumination and data transmission face specific challenges, such as controlling dimming without compromising transmission capacity. In this article, we revisit multipulse pulse-position modulation (MPPM) for joint rate–dimming control in VLC. MPPM is widely considered a highly efficient modulation for optical communications. However, the computational complexity of current MPPM coding methods grows exponentially as the code rate is increased. To overcome this limitation, we propose novel encoding and soft-decision decoding strategies that allow fine-tuning dimming control and increase the data rate. Performance results show higher data rates with lower energy demands as compared to current standards. Moreover, the proposed encoding and decoding methods exhibit linear time complexity, which makes them suitable for implementation in low-cost VLC devices for massive IIoT deployments.

Geometric Constellation Shaping for Concatenated Two-Level Multi-Level Codes

This paper studies a learning approach to geometric shaping (GS) for concatenated two-level multi-level codes (MLC). Unlike standard bit-interleaved coded modulation (BICM), the proposed MLC protects only the least reliable bits (LRBs) by an inner code, and thereby efficiently reduces complexity associated with inner soft-decision (SD) decoding compared to the BICM without performance degradation. We propose a new loss function for training a geometric constellation shape in the proposed MLC, leading to maximization of the achievable rate. More specifically, considering the different coding structures for the most reliable bits (MRBs) and the LRBs, the proposed loss function combines two different functions: the union bound on a bit error rate (BER) for the MRBs and the generalized mutual information (GMI) for the LRBs. We demonstrate by simulations of wavelength division multiplexing (WDM) optical fiber systems that the proposed loss function offers a performance gain over the conventional cross entropy-based loss function. Furthermore, it is demonstrated that the proposed MLC with GS outperforms the conventional BICM with GS in terms of a transmission distance, even with significantly lower SD decoding complexity.

PAPR reduction methods based on bit flipping

ABSTRACT In this work, two peak-to-average-power ratio (PAPR) reduction methods based on bit flipping are designed for OFDM signals. Rather than sending side information, error correction codes are used to recover the flipped bits, as well as to protect the transmitted data. The first PAPR reduction method combines hard-decision decoding (HDD) and natural mapping. Taking advantage of the fact that the natural mapping symbol is represented as a linear combination of the included bits, the flipping bits can be selected at low complexity. The concept of compressed sensing is introduced in order to minimize the number of flipping bits and retain most of the code distance for error correction. In the second method, a low-density parity-check (LDPC) code is cascaded as an inner code following the HDD code, and Gray mapping is used. Soft-decision decoding (SDD) is applied to the LDPC code, which significantly improves the data protection capability, then the outer HDD code is able to focus on recovering the flipped bits. The designs proposed in this paper show great flexibility and provide a trade-off between PAPR reduction and the error-correction capability.

A New Helper Data Scheme for Soft-Decision Decoding of Binary Physical Unclonable Functions

Physical unclonable functions (PUFs) exploit randomness in the hardware for the derivation of cryptographic keys. In the literature, usually the readout is two-level quantized and hard-decision channel decoding is used to stabilize the extracted key. In this paper, we assess soft-decision decoding of binary PUFs. It is well known in the literature on channel coding that soft-decision decoding provides significant gains over hard-decision decoding since reliability information about the symbols is utilized. The PUF readout process is interpreted as digital transmission over a noisy channel, the respective capacity is calculated, and the optimum decoding metric is derived. In addition, we propose an augmented helper data scheme which is suited for soft-decision decoding. This scheme utilizes the fact that operations on the analog readout values are possible, opposed to operations on hard-decided binary symbols in classical PUFs. The security of the new scheme is proven and a possible realization is discussed. The performance is covered by numerical simulations and by applying the scheme to measurement data from FPGA implementations of ring oscillator PUFs.

Fast Simulation of Coded QAM Transmission in White Gaussian Noise at Low Packet Error Rates

Today’s communication system design heavily depends on computer simulation for performance evaluation. The burgeoning ultra-reliable communication systems, however, pose a significant simulation challenge as such systems operate at very low packet error rates (PERs) whereas the required simulation time of the conventional Monte Carlo (MC) method is many times the inverse of the PER. Various importance sampling-type techniques have been developed for more efficient simulation of channel-coded transmission, but they typically rely on exploiting code weaknesses (for the generation of error-causing noise samples) and most works on soft-decision decoding only treat binary signaling. In this paper, we propose to use a function, termed the noise gauging function (NGF), that roughly measures the error-causing propensity of noise samples and we present a way to adaptively optimize the noise sampling under such a function for simulation efficiency. Both binary and nonbinary signalings are considered. And the proposed technique does not require detailed knowledge of the code weaknesses, although some high-level understanding of the code properties can benefit the design of efficient NGFs. We investigate the application of the proposed technique to several common channel codes. Numerical results indicate an approximately 10-to 1,000-fold speedup versus MC.