LDPC Research Articles

Low-overhead fault-tolerant quantum computing using long-range connectivity.

Vast numbers of qubits will be needed for large-scale quantum computing because of the overheads associated with error correction. We present a scheme for low-overhead fault-tolerant quantum computation based on quantum low-density parity-check (LDPC) codes, where long-range interactions enable many logical qubits to be encoded with a modest number of physical qubits. In our approach, logic gates operate via logical Pauli measurements that preserve both the protection of the LDPC codes and the low overheads in terms of the required number of additional qubits. Compared with surface codes with the same code distance, we estimate order-of-magnitude improvements in the overheads for processing around 100 logical qubits using this approach. Given the high thresholds demonstrated by LDPC codes, our estimates suggest that fault-tolerant quantum computation at this scale may be achievable with a few thousand physical qubits at comparable error rates to what is needed for current approaches.

Effective identification of dominant fully absorbing sets for Raptor‐like LDPC codes

AbstractThe error‐rate floor of low‐density parity‐check (LDPC) codes is attributed to the trapping sets of their Tanner graphs. Among them, fully absorbing sets dominantly affect the error‐rate performance, especially for short blocklengths. Efficient methods to identify the dominant trapping sets of LDPC codes were thoroughly researched as exhaustively searching them is NP‐hard. However, the existing methods are ineffective for Raptor‐like LDPC codes, which have many types of trapping sets. An effective method to identify dominant fully absorbing sets of Raptor‐like LDPC codes is proposed. The search space of the proposed algorithm is optimized into the Tanner subgraphs of the codes to afford time‐efficiency and search‐effectiveness. For 5G New Radio (NR) base graph (BG) 2 LDPC codes for short blocklengths, the proposed algorithm finds more dominant fully absorbing sets within one seventh of the computation time of the existing search algorithm, and its search‐effectiveness is verified using importance sampling. The proposed method is also applied to 5G NR BG1 LDPC code and Advanced Television Systems Committee 3.0 type A LDPC code for large blocklengths.

The BER Performance of the LDPC-Coded MPPM over Turbulence UWOC Channels

Turbulence-induced fading is a critical performance degrading factor for underwater wireless optical communication (UWOC) systems. In this paper, we propose a quasi-cyclic (QC) low-density parity-check (LDPC) code with multiple-pulse-position modulation (MPPM) to overcome turbulence-induced fading. MPPM is adopted as a compromise between the low-power efficiency of on–off keying (OOK) and the low bandwidth efficiency of pulse position modulation (PPM). The bit error rate (BER) performance of LDPC-coded MPPM over turbulence UWOC channels is investigated. The log-likelihood ratio (LLR) of MPPM is derived, and a simplified approximation is used for iterative decoding. Subsequently, the closed-form expression of the BER, without forward error correction (FEC) code, is obtained for the generalized-gamma (GG) fading model. Finally, Monte-Carlo (MC) simulation results are provided to demonstrate the correctness of the derived closed-form expressions and the effectiveness of the LDPC code with simplified LLR to improve the BER performance for different MPPM formats over fading channels.

The Effectiveness of LDPC Decoding Algorithms in 5G Channel Modelling of MIMO-OFDM System under the Influence of Spatial Correlation

This paper investigates the spatial cross-correlation properties of 5G channel modeling in non-line of sight (NLOS) of Urban Micro Cell (UMi), Rural Macro Cell (RMa) and Indoor Cell (InH) at 6 GHz frequency band as Third Generation Partnership Project (3GPP) specification. In our spatial correlation Multiple-input multiple-output (MIMO) channel, when increasing the distances of BS antenna elements, the correlation characteristics of the channel dramatically vary, which leads to altering the observed system behavior. By analyzing the 5G LDPC code and using different decoding algorithms and code rates, we evaluate the system’s performance by Bit Error Rate (BER) in a wideband correlated Multiple-input multiple-output Orthogonal Frequency Division Multiplexing (MIMO-OFDM) system. In our spatial correlation, the more rising of the distances of antenna elements on the BS side, the better the effectiveness of the LDPC decoding algorithms achieved. Of the Belief Propagation based (BP-based) Algorithm, Offset Minimum-Sum (OMS) Algorithm and Linear Approximation Minimum-Sum (LAMS) Algorithm, we propose to use the LAMS decoding algorithm with a high code rate of 5/6 in our spatial correlated wideband channel because of lower complexity and more efficiency in term of the system’s performance.

Connectivity constrains quantum codes

Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D-dimensional hyperbolic space.

Anti error and erasure coding for water-to-air visible light communication through wavy water surface with wave height up to 0.6 meters.

Considering large dynamic optical intensity range in a water-to-air (W2A) channel, we propose two promising channel coding schemes, namely the concatenated Reed Solomon-Low Density Parity Check (RS-LDPC) code and Raptor code, for W2A visible light communication (VLC). We establish a W2A-VLC link to verify the performance under different wavy water environments and different water depths with a green light emitting diode (LED). A wave generator is adopted to emulate the wavy water surface with wave height up to 0.6 m. The receiver is fixed 3.2 m above the water, and the transmitter varies from 2.5 m to 4.0 m under the water through a up-down-moveable platform. We test the coding schemes with different code lengths and code rates under 5 MSym/s air-interface symbol rate. Experimental results show that both schemes can reduce the bit error ratio (BER) and frame error rate (FER) of a W2A-VLC system, and thus can improve the reliability. Via comparing the two codes with the same overhead and approximately the same code length, it is demonstrated that Raptor code can generally outperform the concatenated RS-LDPC code. Our research provides promising channel coding methods without feedback for a W2A-VLC system.

Performance analysis of LDPC coded GFDM systems

This paper analyzes the error probability performance of low-density parity-check (LDPC) coded generalized frequency division multiplexing (GFDM) systems over Rayleigh fading and additive white Gaussian noise (AWGN) channels. The initial log-likelihood ratio (LLR) expressions used in the sum-product algorithm (SPA) decoder are first derived for the system model presented in this paper. Based on the decoding threshold of the system, the frame error rate (FER) in the low E b / N 0 $E_b/N_0$ region is estimated by modeling the channel variations using the observed bit error rate (BER). Then, a lower bound based on the absorbing sets is proposed for FER when quantized SPA decoders are used. For AWGN channels, the lower bound can act as an estimate of the FER in the error-floor region if the absorbing set is dominant and its multiplicity is known. For Rayleigh channels, the lower bound can still be used to estimate the FER performance of selected codes. The estimation approach for the FER in the low E b / N 0 $E_b/N_0$ region and the lower bound on the FER in the high E b / N 0 $E_b/N_0$ region can be used as practical tools for evaluating different designs of GFDM-based systems in terms of the error probability performance. The quantization scheme has an important impact on the FER and BER performances. Randomly constructed and array-based LDPC codes are used to obtain numerical results that show the system performance and the accuracy of the proposed FER estimations.

Multi-Mode QC-LDPC Decoding Architecture With Novel Memory Access Scheduling for 5G New-Radio Standard

As the low-density parity-check (LDPC) code has a powerful error-correcting performance and can achieve high throughput, it is being used in many application areas and recently adopted as a channel coding method in the 5G New-Radio communication standard. Unlike other LDPC codes, the 5G LDPC code has various irregular lifting sizes to support diverse message lengths. To meet the demanding requirements of the 5G standard, many solutions have been presented, but all of them are either impractical or fail to satisfy all the requirements. This paper, for the first time, proposes an area-efficient QC-LDPC decoder that satisfies the peak throughput requirements of the 5G standard and supports all the lifting sizes specified in the 5G standard. Instead of relying on full parallelism like in the previous works, this work tries partial parallelism to mitigate the hardware complexity, which leads to high efficiency in hardware complexity. In addition, a novel memory access scheduling method is proposed to solve the data access and alignment problems caused by the partially parallel structure, which is effective in supporting all the lifting sizes. A LDPC decoder realized in 65-nm CMOS technology demonstrates that its decoding throughput is greater than 20Gbps and its area is smaller than the existing decoders.

Deep-Learning for Breaking the Trapping Sets in Low-Density Parity-Check Codes

In the low-error rate regime, message-passing (MP) decoding for low-density parity-check (LDPC) codes is known to have performance degradation due to trapping sets (TSs), which often limits the use of LDPC codes for applications with low target error rates like storage devices. This work proposes a novel deep-learning based decoding algorithm which is tailored for breaking TSs. In particular, when MP decoding fails due to TSs, there exist pairs of unsatisfied check nodes (CNs) which are connected through paths only with error variable nodes (VNs), i.e., VNs with erroneous hard-decision results. The proposed algorithm efficiently identifies the paths with error VNs between unsatisfied CNs with the aid of deep-learning techniques. Then, the decoding failures are resolved by repeating the MP decoding after re-initializing the channel outputs for the error VNs in the identified paths. In addition, by analyzing the behaviors of the deep-learning based algorithm, we propose a low-complexity algorithm, called adaptive-error-path (AEP) detector. Simulation results show that the proposed algorithms efficiently break the TSs and significantly improve the error-floor performance in the low error-rate regime.

An Improved EPA-Based Receiver Design for Uplink LDPC Coded SCMA System

Sparse code multiple access (SCMA) is an emerging paradigm for efficient enabling of massive connectivity in future machine-type communications (MTC). In this letter, we conceive the uplink transmissions of the low-density parity check (LDPC) coded SCMA system. Traditional receiver design of LDPC-SCMA system, which is based on message passing algorithm (MPA) for multiuser detection followed by individual LDPC decoding, may suffer from the drawback of the high complexity and large decoding latency, especially when the system has large codebook size and/or high overloading factor. To address this problem, we introduce a novel receiver design by applying the expectation propagation algorithm (EPA) to the joint detection and decoding (JDD) involving an aggregated factor graph of LDPC code and sparse codebooks. Our numerical results demonstrate the superiority of the proposed EPA based JDD receiver over the conventional Turbo receiver in terms of both significantly lower complexity and faster convergence rate without noticeable error rate performance degradation.

THE PERFORMANCE EVALUATION OF MULTI USER OFDM ORTHOGONAL CHAOTIC VECTOR SHIFT KEYING SUPPORTED BY LDPC

Recently, LDPC code have become very important research area in wireless communication due to its ability to increase the capacity in a wireless fading environment, with low implementation complexity. In this paper, LDPC are combined with Multi User OFDM Orthogonal Chaotic Vector Shift Keying (MU-OFDM-OCVSK) communication system to improve the BER performance over multi-path Rayleigh fading channels. Two types of LDPC decoder are introduced that are Log-Domain and Min-Sum decoder. The system is simulated using MATLAB program version 2019a for different scenarios which include different number of iterations, different block lengths, different number of users and different number of spreading factor. The results show that a coding gain in a range of (4.5 – 7) dB is achieved between the coded and uncoded MU-OFDM-OCVSK system. The results also show that the Min-Sum decoder outperform the Log-Domain decoder in all scenarios.

Design and Analysis of NB QC-LDPC Codes Over Small Alphabets

We propose a novel approach to optimization of irregular nonbinary (NB) quasi-cyclic (QC)-LDPC codes over small alphabets. In this approach, first, the base parity-check matrices are constructed by a simulated annealing method, and then these matrices are labeled by the field elements while maximizing the so- called generalized girth of the Tanner graph. In order to analyze the performance of the constructed irregular NB LDPC codes, a new ensemble of irregular NB LDPC codes over the extensions of the binary Galois field is introduced. A finite-length random coding bound on the error probability of the maximum-likelihood (ML) decoding over the binary phase shift keying (BPSK) input AWGN channel for the new code ensemble is derived. The frame error rate (FER) performance of the sum-product belief-propagation (BP) decoding of the constructed NB QC-LDPC block codes is compared to that of both the optimized binary QC-LDPC block codes in the 5G standard and the best known NB QC-LDPC codes as well as to the derived random coding bound on the ML decoding error probability. It is shown that the obtained bound predicts the behavior of BP decoding performance of practical NB QC-LDPC codes more accurately than the BP decoding thresholds do.

Development of crypto-code constructs based on LDPC codes

The results of developing post-quantum algorithms of McEliece and Niederreiter crypto-code constructs based on LDPC (Low-Density Parity-Check) codes are presented. With the rapid growth of computing capabilities of mobile technologies and the creation of wireless mesh and sensor networks, Internet of Things technologies, and smart technologies on their basis, information security is becoming an urgent problem. At the same time, there is a need to consider security in two circuits, internal (directly within the network infrastructure) and external (cloud technologies). In such conditions, it is necessary to integrate threats to both the internal and external security circuits. This allows you to take into account not only the hybridity and synergy of modern targeted threats, but also the level of significance (degree of secrecy) of information flows and information circulating in both the internal and external security circuits. The concept of building security based on two circuits is proposed. To ensure the security of wireless mobile channels, it is proposed to use McEliece and Niederreiter crypto-code constructs based on LDPC codes, which allows integration into the credibility technology of IEEE 802.15.4, IEEE 802.16 standards. This approach provides the required level of security services (confidentiality, integrity, authenticity) in a full-scale quantum computer. Practical security technologies based on the proposed crypto-code constructs, online IP telephony and the Smart Home system based on the use of an internal server are considered

LDPC codes from cubic semisymmetric graphs

In this paper we study LDPC codes having cubic semisymmetric graphs as their Tanner graphs. We discuss the structure of the smallest absorbing sets of these LDPC codes. Further, we give an expression for the variance of the syndrome weight of the constructed codes, and present computational and simulation results.

Strongly Connected Ramanujan Graphs for Highly Symmetric LDPC Codes

A number of studies focus on Low-Density Parity-Check (LDPC) codes to ensure reliable data communications. This study proposes an algebraic algorithm to generate strongly connected Ramanujan graphs able to provide highly symmetric LDPC codes with minimized error floor. Several Ramanujan graphs are created using GAP system software to generate a rank-efficient parity-check matrix with fixed-rate LDPC codes. We find that Ramanujan LDPC codes achieve frame error rate and bit error rate on the order of \({10}^{-5}\) and \({10}^{-6}\), respectively. Furthermore, the codes outperform QC LDPC codes and those Ramanujan LDPC codes in literature.

Low-Resolution Precoding for Multi-Antenna Downlink Channels and OFDM.

Downlink precoding is considered for multi-path multi-input single-output channels where the base station uses orthogonal frequency-division multiplexing and low-resolution signaling. A quantized coordinate minimization (QCM) algorithm is proposed and its performance is compared to other precoding algorithms including squared infinity-norm relaxation (SQUID), multi-antenna greedy iterative quantization (MAGIQ), and maximum safety margin precoding. MAGIQ and QCM achieve the highest information rates and QCM has the lowest complexity measured in the number of multiplications. The information rates are computed for pilot-aided channel estimation and a blind detector that performs joint data and channel estimation. Bit error rates for a 5G low-density parity-check code confirm the information-theoretic calculations. Simulations with imperfect channel knowledge at the transmitter show that the performance of QCM and SQUID degrades in a similar fashion as zero-forcing precoding with high resolution quantizers.

Memory Efficient Mutual Information-Maximizing Quantized Min-Sum Decoding for Rate-Compatible LDPC Codes

In this letter, we propose a two-stage design method to construct memory efficient mutual information-maximizing quantized min-sum (MIM-QMS) decoder for rate-compatible low-density parity-check (LDPC) codes. We first develop a modified density evolution to design a unique set of lookup tables (LUTs) that can be used for rate-compatible LDPC codes. The constructed LUTs are optimized based on their discrepancy values and a merge function to reduce the memory requirement. Numerical results show that the proposed rate-compatible MIM-QMS decoder can reduce the memory requirement for decoding by up to 94.92 % compared to the benchmark rate-compatible LUT-based decoder with generally faster convergence speed. In addition, the proposed decoder can approach the performance of the floating-pointing belief propagation decoder within 0.15 dB.

Nonbinary I-Polar Coded Modulation

Polar codes were shown to be weak at finite block lengths even under maximum-likelihood (ML) decoding. To enhance polar codes, CRC-aided polar codes were proposed. It has been shown that CRC-aided polar codes under the successive cancellation list (SCL) decoder outperform LDPC codes and turbo codes. Recently, interleaved polar (i-polar) codes were proposed. By using a new bit channel selection algorithm, i-polar codes without CRC were shown to outperform the 5G LDPC codes. However, most research on polar codes focuses on the binary case. In this paper, a new family of nonbinary i-polar codes with random constellation mapping is proposed. A recursive form for calculating the average weight enumerating function (WEF) of the nonbinary i-polar code is derived. With the random constellation mapping, the upper bound on the BLER under ML decoding depends on the WEF of the nonbinary i-polar code. Then a new channel selection algorithm is proposed based on the polarization effect and the BLER upper bound. The algorithm generates a sequence, termed the i-polar sequence (IPS). Simulation results show that the nonbinary i-polar codes with the IPS outperform the 5G LDPC codes.

Reconstruction-Computation-Quantization (RCQ): A Paradigm for Low Bit Width LDPC Decoding

This paper uses the reconstruction-computation-quantization (RCQ)paradigm to decode low-density parity-check (LDPC) codes. RCQ facilitates dynamic non-uniform quantization to achieve good frame error rate (FER) performance with very low message precision. For message-passing according to a flooding schedule, the RCQ parameters are designed by discrete density evolution. Simulation results on an IEEE 802.11 LDPC code show that for 4-bit messages, a flooding Min Sum RCQ decoder outperforms table-lookup approaches such as information bottleneck (IB) or Min-IB decoding, with significantly fewer parameters to be stored. Additionally, this paper introduces layer-specific RCQ, an extension of RCQ decoding for layered architectures. Layer-specific RCQ uses layer-specific message representations to achieve the best possible FER performance. For layer-specific RCQ, this paper proposes using layered discrete density evolution featuring hierarchical dynamic quantization (HDQ) to design parameters efficiently. Finally, this paper studies field-programmable gate array (FPGA) implementations of RCQ decoders. Simulation results for a (9472, 8192) quasi-cyclic (QC) LDPC code show that a layered Min Sum RCQ decoder with 3-bit messages achieves more than a 10% reduction in LUTs and routed nets and more than a 6% decrease in register usage while maintaining comparable decoding performance, compared to a 5-bit offset Min Sum decoder.

A Study on Performance Evaluation With Neural Network Detector in SMR System

We previously applied a neural network detector (NND) to the low-density parity-check (LDPC) coding and iterative decoding system in a shingled magnetic recording (SMR) system. In this study, we evaluate the performance of iterative decoding using a new NND by computer simulation. The conventional NND gets the log-likelihood ratio (LLR) as the reliability of the sequence recorded from the two-dimensional finite impulse response (TD-FIR) filter output and the sum-product (SP) decoder output, but the new NND finds the LLR by adding the SP decoder input in order to consider the difference between the SP decoder input and the decoding result. Furthermore, we clarify that the new NND brings the almost same effect as the a posteriori probability (APP) decoder in the iterative decoding system.