Novel LDPC coding and decoding strategies: design, analysis, and algorithms

In this digital era, modern communication systems play an essential part in nearly every aspect of life, with examples ranging from mobile networks and satellite communications to Internet and data transfer. Unfortunately, all communication systems in a practical setting are noisy, which indicates that we can either improve the physical characteristics of the channel or find a possible systematical solution, i.e. error control coding. The history of error control coding dates back to 1948 when Claude Shannon published his celebrated work “A Mathematical Theory of Communication”, which built a framework for channel coding, source coding and information theory. For the first time, we saw evidence for the existence of channel codes, which enable reliable communication as long as the information rate of the code does not surpass the so-called channel capacity. Nevertheless, in the following 60 years none of the codes have been proven closely to approach the theoretical bound until the arrival of turbo codes and the renaissance of LDPC codes. As a strong contender of turbo codes, the advantages of LDPC codes include parallel implementation of decoding algorithms and, more crucially, graphical construction of codes. However, there are also some drawbacks to LDPC codes, e.g. significant performance degradation due to the presence of short cycles or very high decoding latency. In this thesis, we will focus on the practical realisation of finite-length LDPC codes and devise algorithms to tackle those issues. Firstly, rate-compatible (RC) LDPC codes with short/moderate block lengths are investigated on the basis of optimising the graphical structure of the tanner graph (TG), in order to achieve a variety of code rates (0.1 < R < 0.9) by only using a single encoder-decoder pair. As is widely recognised in the literature, the presence of short cycles considerably reduces the overall performance of LDPC codes which significantly limits their application in communication systems. To reduce the impact of short cycles effectively for different code rates, algorithms for counting short cycles and a graph-related metric called Extrinsic Message Degree (EMD) are applied with the development of the proposed puncturing and extension techniques. A complete set of simulations are carried out to demonstrate that the proposed RC designs can largely minimise the performance loss caused by puncturing or extension. Secondly, at the decoding end, we study novel decoding strategies which compensate for the negative effect of short cycles by reweighting part of the extrinsic messages exchanged between the nodes of a TG. The proposed reweighted belief propagation (BP) algorithms aim to implement efficient decoding, i.e. accurate signal reconstruction and low decoding latency, for LDPC codes via various design methods. A variable factor appearance probability belief propagation (VFAP-BP) algorithm is proposed along with an improved version called a locally-optimized reweighted (LOW)-BP algorithm, both of which can be employed to enhance decoding performance significantly for regular and irregular LDPC codes. More importantly, the optimisation of reweighting parameters only takes place in an offline stage so that no additional computational complexity is required during the real-time decoding process. Lastly, two iterative detection and decoding (IDD) receivers are presented for multiple-input multiple-output (MIMO) systems operating in a spatial multiplexing configuration. QR decomposition (QRD)-type IDD receivers utilise the proposed multiple-feedback (MF)-QRD or variable-M (VM)-QRD detection algorithm with a standard BP decoding algorithm, while knowledge-aided (KA)-type receivers are equipped with a simple soft parallel interference cancellation (PIC) detector and the proposed reweighted BP decoders. In the uncoded scenario, the proposed MF-QRD and VM-QRD algorithms are shown to approach optimal performance, yet require a reduced computational complexity. In the LDPC-coded scenario, simulation results have illustrated that the proposed QRD-type IDD receivers can offer near-optimal performance after a small number of detection/decoding iterations and the proposed KA-type IDD receivers significantly outperform receivers using alternative decoding algorithms, while requiring similar decoding complexity

Fifteen years of quantum LDPC coding and improved decoding strategies

The near-capacity performance of classical low-density parity check (LDPC) codes and their efficient iterative decoding makes quantum LDPC (QLPDC) codes a promising candidate for quantum error correction. In this paper, we present a comprehensive survey of QLDPC codes from the perspective of code design as well as in terms of their decoding algorithms. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank-Shor-Steane-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-four cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their word error rate performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic uniformly reweighted belief propagation for the sake of achieving an improved performance

Stabilizer Inactivation for Message-Passing Decoding of Quantum LDPC Codes

We propose a post-processing method for message-passing (MP) decoding of CSS quantum LDPC codes, called stabilizer-inactivation (SI). It relies on inactivating a set of qubits, supporting a check in the dual code, and then running the MP decoding again. This allows MP decoding to converge outside the inactivated set of qubits, while the error on these is determined by solving a small, constant size, linear system. Compared to the state of the art post-processing method based on ordered statistics decoding (OSD), we show through numerical simulations that MP-SI outperforms MP-OSD for different quantum LDPC code constructions, different MP decoding algorithms, and different MP scheduling strategies, while having a significantly reduced complexity

Coding and Probabilistic Inference Methods for Data-Dependent Two-Dimensional Channels

Recent advances in magnetic recording systems, optical recording devices and flash memory drives necessitate to study two-dimensional (2-D) coding techniques for reliable storage/retrieval of information. Most channels in such systems introduce errors in messages in response to certain data patterns, and messages containing these patterns are more prone to errors than others. For example, in a single-level cell flash memory channel, inter-cell interference (ICI) is at its maximum when 101 patterns are programmed over adjacent cells in either horizontal or vertical directions. As another example, in two-dimensional magnetic recording channels, 2-D isolated-bits patterns are shown empirically to be the dominant error event, and during the read-back process inter-symbol interference (ISI) and inter-track interference (ITI) arise when these patterns are recorded over the magnetic medium. Shannon in his seminal work, ``A Mathematical Theory of Communications," presented two techniques for reliable transmission of messages over noisy channels, namely error correction coding and constrained coding. In the first method, messages are protected via an error correction code (ECC) from random errors which are independent of input data. The theory of ECCs is well studied, and efficient code construction methods are developed for simple binary channels, additive white Gaussian noise (AWGN) channels and partial response channels. On the other hand, constrained coding reduces the likelihood of corruption by removing problematic patterns before transmission over data-dependent channels. Prominent examples of constraints include a family of binary one-dimensional (1-D) and 2-D $\left(d,k\right)$-run-length-limited (RLL) constraints which improves resilience to ISI timing recovery and synchronization for bandwidth limited partial response channels, where d and k represent the minimum and maximum number of admissible zeros between two successive ones in any direction of array. In principle, the ultimate coding approach for such data-dependent channels is to design a set of sufficiently distinct error correction codewords that also satisfy channel constraints. Designing channel codewords satisfying both ECC and channel constraints is important as it would achieve the channel capacity. However, in practice this is difficult, and we rely on sub-optimal methods such as forward concatenation method (standard concatenation), reverse concatenation method (modified concatenation), and combinations of these approaches. In this dissertation, we focus on the problem of reliable transmission of binary messages over data-dependent 2-D communication channels. Our work is concerned with several challenges in regard to the transmission of binary messages over data-dependent 2-D channels. Design of Two-Dimensional Magnetic Recording (TDMR) Detector and Decoder: TDMR achieves high areal densities by reducing the size of a bit comparable to the size of the magnetic grains resulting in 2-D ISI and very high media noise. Therefore, it is critical to handle the media noise along with the 2-D ISI detection. In this work, we tune the Generalized Belief Propagation (GBP) algorithm to handle the media noise seen in TDMR. We also provide an intuition into the nature of hard decisions provided by the GBP algorithm. Investigation into Harmful Patterns for TDMR channels: This work investigates into the Voronoi based media model to study the harmful patterns over multi-track shingled recording systems. Through realistic quasi micromagnetic simulations studies, we identify 2-D data patterns that contribute to high media noise. We look into the generic Voronoi model and present our analysis on multi-track detection with constrained coded data. We show that 2-D constraints imposed on input patterns result in an order of magnitude improvement in the bit error rate for TDMR systems. Understanding of Constraint Gain for TDMR Channels: We study performance gains of constrained codes in TDMR channels using the notion of constraint gain. We consider Voronoi based TDMR channels with realistic grain, bit, track and magnetic-head dimensions. Specifically, we investigate the constraint gain for 2-D no-isolated-bits constraint over Voronoi based TDMR channels. We focus on schemes that employ the GBP algorithm for obtaining information rate estimates for TDMR channels. Design of Novel Constrained Coding Methods: In this work, we present a deliberate bit flipping (DBF) coding scheme for binary 2-D channels, where specific patterns in channel inputs are the significant cause of errors. The idea is to eliminate a constrained encoder and, instead, embed a constraint into an error correction codeword that is arranged into a 2-D array by deliberately flipping the bits that violate the constraint. The DBF method relies on the error correction capability of the code being used so that it should be able to correct both deliberate errors and channel errors. Therefore, it is crucial to flip minimum number of bits in order not to overburden the error correction decoder. We devise a constrained combinatorial formulation for minimizing the number of flipped bits for a given set of harmful patterns. The GBP algorithm is used to find an approximate solution for the problem. Devising Reduced Complexity Probabilistic Inference Methods: We propose a reduced complexity GBP that propagates messages in Log-Likelihood Ratio (LLR) domain. The key novelties of the proposed LLR-GBP are: (i) reduced fixed point precision for messages instead of computational complex floating point format, (ii) operations performed in logarithm domain, thus eliminating the need for multiplications and divisions, (iii) usage of message ratios that leads to simple hard decision mechanisms

On The Analysis of Spatially-Coupled GLDPC Codes and The Weighted Min-Sum Algorithm

This dissertation studies methods to achieve reliable communication over unreliable channels. Iterative decoding algorithms for low-density parity-check (LDPC) codes and generalized LDPC (GLDPC) codes are analyzed. A new class of error-correcting codes to enhance the reliability of the communication for high-speed systems, such as optical communication systems, is proposed. The class of spatially-coupled GLDPC codes is studied, and a new iterative hard- decision decoding (HDD) algorithm for GLDPC codes is introduced. The main result is that the minimal redundancy allowed by Shannon’s Channel Coding Theorem can be achieved by using the new iterative HDD algorithm with spatially-coupled GLDPC codes. A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with iterative decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. To the best of our knowledge, this is the first system that can approach the channel capacity using iterative HDD. The optimality of a codeword returned by the weighted min-sum (WMS) algorithm, an iterative decoding algorithm which is widely used in practice, is studied as well. The attenuated max-product (AttMP) decoding and weighted min-sum (WMS) decoding for LDPC codes are analyzed. Applying the max-product (and belief- propagation) algorithms to loopy graphs are now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This work presents an analysis of both AttMP decoding and WMS decoding for LDPC codes which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution