Low-density Generator Matrix Research Articles

Application of the method of equivalent message encoding structures to simplify coding complexity

Formulation of the problem. Modern telecommunication technologies make it possible to transmit data over long distances with limitations in the form of low power consumption and low error rate using error correction codes. The use of such codes plays an important role and is necessary to detect and correct errors caused by signal distortion and noise in the communication channel. According to Shannon's channel coding theorem, the probability of error can be arbitrarily small or close to zero if the encoding rate of transmission R is less than or equal to the bandwidth of channel C. Assuming that the bandwidth C is achievable, we can calculate the minimum signal-to-noise power ratio (SNR) to achieve zero error, or an arbitrarily small error called the Shannon limit, while maintaining R ≤ C. This article discusses a method of applying equivalent message encoding structures to construct low-density parity-checking (LDPC) codes. Low-density parity-checking (LDPC) codes are a class of linear block codes in which the complexity of iterative decoding increases linearly with increasing block length. Standard LDPC codes without modifications are considered to demonstrate the effectiveness of the proposed method, which is expected to provide high encoding speed when transmitting messages with a short block length. The method of obtaining equivalent structures allows: to reduce the computational complexity of encoding and decoding; to provide the possibility of finding a new encoding scheme and evaluating its effectiveness. The validity of the method for obtaining equivalent message encoding structures is evaluated by modeling the bit error rate (BER) of LDPC codes. Purpose. to show the effectiveness of the use of equivalent structures for various methods of compression of transmitted information, to minimize the complexity of encoding, while maintaining energy efficiency, correcting ability and data transfer rate. Results. A method is considered for obtaining equivalent message element encoding structures that can be used to create more LDPC codes for data transmission applications with a short block length. The cascade principle of constructing an equivalent generator and a parity check matrix was analyzed, which also leads to minimizing the computational complexity of the encoder and decoder. An equivalent structure G can be obtained from cascading two (or more) matrices by modifying the second matrix so that it has a unit matrix with the size of the parity check length and a shift to the right by k bits. On the other hand, the equivalent structure H can be obtained by simply cascading directly the second matrix and shifting to the right by k bits. According to the simulation results, the proposed method is adequate, since LDPC codes have the same BER characteristics as codes based on the low density generator matrix (LDGM). During the simulation, it was concluded that the SPC codes (single parity check (single parity check (SPC)) are the most efficient codes (for a given block length), since SPC codes have the smallest interval up to the Shannon limit. Practical significance. The results presented in this article will give a new insight into the development of simple channel coding of LDPC codes of short block length with high encoding speed for future applications with lower power consumption.

Deriving equivalent structure of elements for low density parity check codes construction

This paper proposes a method to derive equivalent coding structures of elements to construct low density parity check (LDPC) codes. We propose stairs LDPC (SLDPC) codes to demonstrate the effectiveness of the proposed method, which is expected to be beneficial for short block-length transmissions, but providing high coding rate. The equivalent coding structures are both for transmitter and receiver to: (i) reduce the encoding and decoding computational complexity, and (ii) search possibility of finding new coding scheme and observe their performances. We evaluate the validity of the method by confirming the equality in performances of the SLDPC codes in terms of bit-error-rate (BER) followed by investigation on their performance gaps to the Shannon limit via a series of computer simulations. The results show that the SLDPC codes have the same BER performance with that of the low density generator matrix (LDGM) codes confirming the validity of the proposed equivalent matrix derivation. This result indicates that different graphs can provide the same performances, because their equivalent matrices are the same. This result is expected to open new insight for the designing simple channel coding for short block-length LDPC codes having high coding rate for future less power consumption applications.

Systematic Convolutional Low Density Generator Matrix Code

In this paper, we propose a systematic low density generator matrix (LDGM) code ensemble, which is defined by the Bernoulli process. We prove that, under maximum likelihood (ML) decoding, the proposed ensemble can achieve the capacity of binary-input output symmetric (BIOS) memoryless channels in terms of bit error rate (BER). The proof technique reveals a new mechanism, different from lowering down frame error rate (FER), that the BER can be lowered down by assigning light codeword vectors to light information vectors. The finite length performance is analyzed by deriving an upper bound and a lower bound, both of which are shown to be tight in the high signal-to-noise ratio (SNR) region. To improve the waterfall performance, we construct the systematic convolutional LDGM (SysConv-LDGM) codes by a random splitting process. The SysConv-LDGM codes are easily configurable in the sense that any rational code rate can be realized without complex optimization. As a universal construction, the main advantage of the SysConv-LDGM codes is their near-capacity performance in the waterfall region and predictable performance in the error-floor region that can be lowered down to any target as required by increasing the density of the uncoupled LDGM codes. Numerical results are also provided to verify our analysis.

Errors and Erasures Decoding of Product Codes for Optical Transport Networks

A concatenated soft decision forward error correcting (FEC) coding scheme with an inner low density generator matrix (LDGM) code and an outer product code (PC) is considered for high-speed optical communications. We compare the performance of the concatenated coding scheme in which erasures are introduced at the output of the inner decoder to that of the scheme in which hard-decision is made at the input of inner decoder. An exact expression for the number of errors and erasures that are guaranteed to be decoded by the outer product code is derived. Then, an approach for deriving an approximation of the error probability for the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">erasure</i> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">error</i> decoder is proposed. Using this approach, optimal thresholds can be chosen for declaring erasures at the output of the inner decoder. It is shown that the codeword error rate for the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">error</i> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">erasure</i> iterative decoder is better than that of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">error only</i> iterative decoder in the error floor region.

Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

Quantum low-density generator matrix (QLDGM) codes based on Calderbank-Steane-Shor (CSS) constructions have shown unprecedented error correction capabilities, displaying much improved performance in comparison to other sparse-graph codes. However, the nature of CSS designs and the manner in which they must be decoded limit the performance that is attainable with codes that are based on this construction. This motivates the search for quantum code design strategies capable of avoiding the drawbacks associated with CSS codes. In this article, we introduce non-CSS quantum code constructions based on classical LDGM codes. The proposed codes are derived from CSS QLDGM designs by performing specific row operations on their quantum parity check matrices to modify the associated decoding graphs. The application of this method results in performance improvements in comparison to CSS QLDGM codes, while also allowing for greater flexibility in the design process. The proposed non-CSS QLDGM scheme outperforms the best quantum low-density parity check codes that have appeared in the literature.

3D EXIT Charts for Analyzing the 5G 3GPP New Radio LDPC Decoder

Low Density Parity Check (LDPC) codes have been selected to provide channel coding for data in the next generation of wireless communication standard referred to in practice as the 3GPP New Radio (NR). In contrast to the LDPC codes adopted in previous standards, the NR LDPC code can be considered to be a concatenation of a core LDPC code and a Low Density Generator Matrix (LDGM) code. This particular feature has the advantage of offering flexible coding rate, but it prevents the application of conventional extrinsic information transfer (EXIT) chart analysis. This article characterizes this problem and addresses it using a novel reinterpretation of the NR LDPC factor graph. Based on this factor graph, a novel 3D EXIT chart technique is conceived for our three-stage scheme, which facilitates the visual characterisation of the NR LDPC decoder's iterative decoding convergence process for the first time. The proposed EXIT chart analysis accurately visualizes the mutual information exchange amongst the components of the NR LDPC decoder, which was not facilitated by the conventional 2D EXIT chart. We demonstrate the power of this technique by using it to design a novel iterative decoding activation order for the NR LDPC decoder, which reduces the decoding complexity by approximately 17% compared to a conventional flooding decoder. This is achieved without degrading its error correction capability. We conclude by discussing several other opportunities for exploiting the proposed 3D EXIT chart technique to improve the design of concatenated LDPC and LDGM codes.

Successive Wyner-Ziv Coding for the Binary CEO Problem Under Logarithmic Loss

The $L$-link binary Chief Executive Officer (CEO) problem under logarithmic loss is investigated in this paper. A quantization splitting technique is applied to convert the problem under consideration to a $(2L-1)$-step successive Wyner-Ziv (WZ) problem, for which a practical coding scheme is proposed. In the proposed scheme, low-density generator-matrix (LDGM) codes are used for binary quantization while low-density parity-check (LDPC) codes are used for syndrome generation; the decoder performs successive decoding based on the received syndromes and produces a soft reconstruction of the remote source. The simulation results indicate that the rate-distortion performance of the proposed scheme can approach the theoretical inner bound based on binary-symmetric test-channel models.

Combining the Burrows-Wheeler Transform and RCM-LDGM Codes for the Transmission of Sources with Memory at High Spectral Efficiencies.

In this paper, we look at the problem of implementing high-throughput Joint Source- Channel (JSC) coding schemes for the transmission of binary sources with memory over AWGN channels. The sources are modeled either by a Markov chain (MC) or a hidden Markov model (HMM). We propose a coding scheme based on the Burrows-Wheeler Transform (BWT) and the parallel concatenation of Rate-Compatible Modulation and Low-Density Generator Matrix (RCM-LDGM) codes. The proposed scheme uses the BWT to convert the original source with memory into a set of independent non-uniform Discrete Memoryless (DMS) binary sources, which are then separately encoded, with optimal rates, using RCM-LDGM codes.

Asymptotic BER EXIT chart analysis for high rate codes based on the parallel concatenation of analog RCM and digital LDGM codes

This paper proposes an extrinsic information transfer (EXIT) chart analysis and an asymptotic bit error rate (BER) prediction method to speed up the design of high rate RCM-LDGM hybrid codes over AWGN and fast Rayleigh channels. These codes are based on a parallel concatenation of a rate compatible modulation (RCM) code with a low-density generator matrix (LDGM) code. The decoder uses the iterative sum-product algorithm to exchange information between the variable nodes (VNs) and the two types of constituent check nodes: RCM-CN and LDGM-CN. The novelty of the proposed EXIT chart procedure lies on the fact that it mixes together the analog RCM check nodes with the digital LDGM check nodes, something not possible in previous multi-edge EXIT charts proposed in the literature.

On Privacy Amplification, Lossy Compression, and Their Duality to Channel Coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite-blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the $E_\gamma$ divergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with known relations between channel simulation and lossy source coding, this implies that privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Info. Theory 64, 577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a "curious duality" in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.

Encoding of Spatially Coupled LDGM Codes for Lossy Source Compression

It has been shown that a class of spatially coupled low-density generator-matrix (SC-LDGM) code ensembles displays distortion saturation for the lossy binary symmetric source coding problem with the belief propagation guided decimation (BPGD) algorithm, i.e., the BPGD distortion approaches the optimal expected distortion of the underlying ensemble asymptotically in code length. We investigate the distortion performance of a practical class of protograph-based SC-LDGM code ensembles and demonstrate distortion saturation numerically. Moreover, taking advantage of the convolutional structure of the SC-LDGM codes, we propose an efficient windowed encoding (WE) algorithm with two decimation techniques for lowering the WE complexity that maintain distortion performance close to the rate-distortion bound.

Information-theoretic thresholds from the cavity method

Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models. We work out several concrete applications of these general results. For instance, we pinpoint the exact condensation phase transition in the Potts antiferromagnet on the random graph, thereby improving prior approximate results (Contucci et al., 2013) [34]. Further, we prove the conjecture from Krzakala et al. (2007) [55] about the condensation phase transition in the random graph coloring problem for any number q≥3 of colors. Moreover, we prove the conjecture on the information-theoretic threshold in the disassortative stochastic block model (Decelle et al., 2011) [35]. Additionally, our general result implies the conjectured formula for the mutual information in Low-Density Generator Matrix codes (Montanari, 2005) [73].

Flexible WOM codes for NAND flash memory based on raptor-like codes

Write-once memory (WOM) codes aim to extend the lifetime and improve the writing efficiency of storage devices such as NAND flash memory by reducing the number of erase operations. In this paper, a new rewriting scheme for NAND flash memory is proposed, which supports two writes (or only one rewrite) and allows the second write incrementally done multiple times by using raptor-like codes as rate-compatible (RC) low-density generator matrix (LDGM) codes. The proposed scheme improves writing efficiency of the NAND flash memory when combined with a proper page selection method. It is verified via numerical analysis that the proposed WOM codes outperform the conventional WOM codes in terms of writing efficiency.

Randomized Serially Concatenated LDGM Codes for the Gaussian Wiretap Channel

We study the application of a special class of low-density parity-check codes to the wiretap channel. We construct a randomized coding scheme based on serially concatenated low-density generator matrix codes and their duals extending the approach used for convolutional and turbo codes. Furthermore, we propose an efficient iterative decoder for this scheme utilizing a joint iterative message passing algorithm. We demonstrate via numerical examples that this approach outperforms other available practical coding alternatives for the Gaussian wiretap channel in terms of the resulting security gap.

Binary Wyner–Ziv code design based on compound LDGM–LDPC structures

In this study, a practical coding scheme is designed for the binary Wyner–Ziv (WZ) problem by using nested low-density generator-matrix (LDGM) and low-density parity-check (LDPC) codes. This scheme contains two steps in the encoding procedure. The first step involves applying the binary quantisation by employing LDGM codes and the second one is using the syndrome-coding technique by utilising LDPC codes. The decoding algorithm of the proposed scheme is based on the sum-product algorithm with the help of a side information available at the decoder side. It is theoretically shown that the compound structure has the capability of achieving the WZ bound. The proposed method approaches this bound by utilising the iterative message-passing algorithms in both encoding and decoding, although theoretical results show that it is asymptotically achievable.

Toward Practical Code-Based Signature: Implementing Fast and Compact QC-LDGM Signature Scheme on Embedded Hardware

In this paper, fast and compact implementations for code-based signature are presented. Existing designs are either using enormous memory storage or suffering from slow issuing speed of signatures. A vastly optimized new design solving these problems is proposed by exploiting quasi-cyclic low-density generator matrix codes at different levels. In particular, this paper provides a new algorithmic enhancement of signature generation and gives detailed and optimized solutions for critical steps of this algorithm. The design presented in this paper is the fastest implementation of code-based signatures in open literature. It is shown, for instance, that our implementation of signature generation engine can generate approximately 60 000 signatures per second on a Xilinx Virtex-6 FPGA, requiring only 5992 slices and 60 memory blocks. In addition, a very compact implementation is also provided, producing 5438 signatures per second with only 18 memory blocks.

Low-Complexity Soft-Decision Concatenated LDGM-Staircase FEC for High-Bit-Rate Fiber-Optic Communication

A concatenated soft-decision forward error correction (FEC) scheme consisting of an inner low-density generator-matrix (LDGM) code and an outer staircase code is proposed. The soft-decision LDGM code is used for error reduction, while the majority of bit errors are corrected by the low-complexity hard-decision staircase code. Decoding complexity of the concatenated code is quantified by a score based on the number of edges in the LDGM code Tanner graph, the number of decoding iterations, and the number of staircase code decoding operations. The inner LDGM ensemble is designed by solving an optimization problem, which minimizes the product of the average node degree and an estimate of the required number of decoding iterations. A search procedure is used to find the inner and outer code pair with lowest complexity. The design procedure results in a Pareto-frontier characterization of the tradeoff between net coding gain and complexity for the concatenated code. Simulations of code designs at $\text{20}\%$ overhead showed that the proposed scheme achieves net coding gains equivalent to existing soft-decision FEC solutions, with up to $\text{57}\%$ reduction in complexity.

Bounds on the Reliability of RaptorQ Codes in the Finite-Length Regime

In this paper we analyze the bounds on the reliability of RaptorQ codes under maximum likelihood decoding, especially in the finite-length regime. RaptorQ code ensembles by a high-order low density generator-matrix (LDGM) code as pre-code and an inner Luby transform (LT) code. By investigating the rank of the product of two random coefficient matrices of the high-order LDGM code and the inner LT code, we derive the expressions for the upper and lower bounds of decoding failure probability (DFP) on the RaptorQ code. Then, the accuracy of our derived theoretical bounds are verified through the Monte Carlo simulations with different degree distributions and short packets. The high accuracy bounds are then exploited to design near-optimum finite-length RaptorQ codes, enabling a tight control on the tradeoff between decoding complexity and DFP.

Video over DSL with LDGM Codes for Interactive Applications

Digital Subscriber Line (DSL) network access is subject to error bursts, which, for interactive video, can introduce unacceptable latencies if video packets need to be re-sent. If the video packets are protected against errors with Forward Error Correction (FEC), calculation of the application-layer channel codes themselves may also introduce additional latency. This paper proposes Low-Density Generator Matrix (LDGM) codes rather than other popular codes because they are more suitable for interactive video streaming, not only for their computational simplicity but also for their licensing advantage. The paper demonstrates that a reduction of up to 4 dB in video distortion is achievable with LDGM Application Layer (AL) FEC. In addition, an extension to the LDGM scheme is demonstrated, which works by rearranging the columns of the parity check matrix so as to make it even more resilient to burst errors. Telemedicine and video conferencing are typical target applications.

High-performance forward error correction: Enabling multi-gigabit flows and beyond on commodity GPU and CPU hardware in presence of packet loss

In demanding real-time multimedia transmissions, even a small packet loss might significantly degrade the visual quality. As retransmission is not an option in real-time transfers especially when transmitting the data over long distances, it is necessary to employ mechanisms of Forward Error Correction (FEC). Low-Density Generator Matrix (LDGM) codes are known to be suitable for coding on large block sizes, however, high bitrates of currently used video formats (FullHD, 4K) also require high throughput of FEC coding and decoding. We propose a parallel design of LDGM encoding and decoding algorithms suitable for off-the-shelf, (massively) parallel platforms, such CPUs with vector units or GPUs, and evaluate our approach in real-world scenarios of high-definition and 4K video transmissions. Our results show that offloading FEC computation to such platform is beneficial for low-latency, high-quality multimedia transmissions and may even enable transmissions beyond 10Gbps once the commodity network interfaces reach this speed.