We investigate iterative algebraic decoding of codes defined on graphs. Practical codes that can achieve the Shannon limit are codes defined on graphs with its associated iterative decoding algorithms. Yet, the complexity of these algorithms is high, especially for long codes that attain capacity. For the foreseeable future, algebraic decoding will still be a standard in industry. We aim to harness the power of iterative processes and the low complexity of algebraic techniques by decoding codes on graphs using iterative algebraic techniques. We study the performance of codes on graphs decoded with this method taking into account the case of undetected errors. More specifically, we examine the threshold under which decoding is successful for the BEC and the BSC and compare these with that of certain LDPC codes. Our initial investigation shows that although a significant loss of performance is incurred on the BEC when compared to belief propagation decoding, for transmission over the BSC the threshold value is somewhat close to belief propagation. Especially for high rate codes iterative algebraic decoding could have good performance while maintaining low decoding complexity.
We investigate iterative algebraic decoding of codes defined on graphs. Practical codes that can achieve the Shannon limit are codes defined on graphs with its associated iterative decoding algorithms. Yet, the complexity of these algorithms is high, especially for long codes that attain capacity. For the foreseeable future, algebraic decoding will still be a standard in industry. We aim to harness the power of iterative processes and the low complexity of algebraic techniques by decoding codes on graphs using iterative algebraic techniques. We study the performance of codes on graphs decoded with this method taking into account the case of undetected errors. More specifically, we examine the threshold under which decoding is successful for the BEC and the BSC and compare these with that of certain LDPC codes. Our initial investigation shows that although a significant loss of performance is incurred on the BEC when compared to belief propagation decoding, for transmission over the BSC the threshold value is somewhat close to belief propagation. Especially for high rate codes iterative algebraic decoding could have good performance while maintaining low decoding complexity.
The growing popularity of a class of linear block codes called the low-density parity-check (LDPC) codes can be attributed to the low complexity of the iterative decoders, and their potential to achieve performance very close to the Shannon capacity. This makes them an attractive candidate for ECC applications in communication systems. This report proposes methods to systematically construct regular and irregular LDPC codes.A class of regular LDPC codes are constructed from incidence structures in finite geometries like projective geometry and affine geometry. A class of irregular LDPC codes are constructed by systematically splitting blocks of balanced incomplete block designs to achieve desired weight distributions. These codes are decoded iteratively using message-passing algorithms, and the performance of these codes for various channels are presented in this report.The application of iterative decoders is generally limited to a class of codes whose graph representations are free of small cycles. Unfortunately, the large class of conventional algebraic codes, like RS codes, has several four cycles in their graph representations. This report proposes an algorithm that aims to alleviate this drawback by constructing an equivalent graph representation that is free of four cycles. It is theoretically shown that the four-cycle free representation is better suited to iterative erasure decoding than the conventional representation. Also, the new representation is exploited to realize, with limited success, iterative decoding of Reed-Solomon codes over the additive white Gaussian noise channel.Wiberg, Forney, Richardson, Koetter, and Vontobel have made significant contributions in developing theoretical frameworks that facilitate finite length analysis of codes. With an exception of Richardson's, most of the other frameworks are much suited for the analysis of short codes. In this report, we further the understanding of the failures in iterative decoders for the binary symmetric channel. The failures of the decoder are classified into two categories by defining trapping sets and propagating sets. Such a classification leads to a successful estimation of the performance of codes under the Gallager B decoder. Especially, the estimation techniques show great promise in the high signal-to-noise ratio regime where the simulation techniques are less feasible.
Low-density parity-check (LDPC) codes have been known for their outstanding error-correction capabilities. With low- complexity decoding algorithms and a near capacity performance, these codes are among the most promising forward error correction schemes. LDPC decoding algorithms are generally sub-optimal and their performance not only depends on the codes, but also on many other factors, such as the code representation. In particular, a given non- binary code can be associated with a number of different field or ring image codes. Additionally, each LDPC code can be described with many different Tanner graphs. Each of these different images and graphs can possibly lead to a different performance when used with iterative decoding algorithms. Consequently, in this dissertation we try to find better representations, i.e., graphs and images, for LDPC codes. We take the first step by analyzing LDPC codes over multiple-input single-output (MISO) channels. In an n_T by 1 MISO system with a modulation of alphabet size 2M̂, each group of n_T transmitted symbols are combined and produce one received symbol at the receiver. As a result, we consider the LDPC-coded MISO system as an LDPC code over a 2̂{M n_T}-ary alphabet. We introduce a modified Tanner graph to represent MISO-LDPC systems and merge the MISO symbol detection and binary LDPC decoding steps into a single message passing decoding algorithm. We present an efficient implementation for belief propagation decoding that significantly reduces the decoding complexity. With numerical simulations, we show that belief propagation decoding over modified graphs outperforms the conventional decoding algorithm for short length LDPC codes over unknown channels. Subsequently, we continue by studying images of non-binary LDPC codes. The high complexity of belief propagation decoding has been proven to be a detrimental factor for these codes. Thereby, we suggest employing lower complexity decoding algorithms over image codes instead. We introduce three classes of binary image codes for a given non-binary code, namely: basic, mixed, and extended binary image codes. We establish upper and lower bounds on the minimum distance of these binary image codes, and present two techniques to find binary image codes with better performance under belief propagation decoding algorithm. In particular, we present a greedy algorithm to find optimized binary image codes. We then proceed by investigation of the ring image codes. Specifically, we introduce matrix-ring-image codes for a given non-binary code. We derive a belief propagation decoding algorithm for these codes, and with numerical simulations, we demonstrate that the low-complexity belief propagation decoding of optimized image codes has a performance very close to the high complexity BP decoding of the original non-binary code. Finally, in a separate study, we investigate the performance of iterative decoders over binary erasure channels. In particular, we present a novel approach to evaluate the inherent unequal error protection properties of irregular LDPC codes over binary erasure channels. Exploiting the finite length scaling methodology, that has been used to study the average bit error rate of finite-length LDPC codes, we introduce a scaling approach to approximate the bit erasure rates in the waterfall region of variable nodes with different degrees. Comparing the bit erasure rates obtained from Monte Carlo simulation with the proposed scaling approximations, we demonstrate that the scaling approach provides a close approximation for a wide range of code lengths. In view of the complexity associated with the numerical evaluation of the scaling approximation, we also derive simpler upper and lower bounds and demonstrate through numerical simulations that these bounds are very close to the scaling approximation
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.
At the heart of modern coding theory lies the fact that low-density parity-check (LDPC) codes can be efficiently decoded by belief propagation (BP). The BP is an inference algorithm which operates on a graphical model of a code, and lends itself to low-complexity and high-speed implementations, making it the algorithm of choice in many applications. It has unprecedentedly good error rate performance, so good that when decoded by the BP, LDPC codes approach theoretical limits of channel capacity. However, this capacity approaching property holds only in the asymptotic limit of code length, while codes of practical lengths suffer abrupt performance degradation in the low noise regime known as the error floor phenomenon. Our study of error floor has led to an interesting and surprising finding that it is possible to design iterative decoders which are much simpler yet better than belief propagation! These decoders do not propagate beliefs but a rather different kind of messages that reflect the local structure of the code graph. This has opened a plethora of exciting theoretical problems and applications. This paper introduces this new paradigm.
Efficient decoding is crucial to high-throughput and power-sensitive wireless communication scenarios. A theoretical analysis of the performance-complexity tradeoff toward low-complexity decoding is required for a better understanding of the fundamental limits in the above-mentioned scenarios. This study aims to explore the performance of LDPC codes under belief propagation (BP) decoding with complexity constraints. In other words, for a small number of iterations, we present a closed-form lower bound on the bit error rate (BER) of LDPC codes as a function of complexity. Specifically, for the regular LDPC code ensembles, the dominant term in the order of the lower bound we provide matches that of the upper bound given by Lentmaier. Furthermore, for irregular LDPC code ensembles, in addition to adopting the approach used for regular codes, we also propose a method to iteratively obtain the lower bound on the average BER.
Since the invention of turbo codes, there has been an explosion of interest in iterative decoding and graphical representation of codes. This thesis examines the iterative decoding of codes defined on graphs with cycles, which appears to be an efficient means of achieving the Shannon limit. Much of this analysis is on the iterative min-sum decoding of tail-biting codes and cycle codes. We have identified the pseudocodeword as the cause of the suboptimal performance of the iterative decoder, and we have obtained a union bound for the performance of the iterative decoder on both AWGN and BSC channels. Using the union bound argument, for cycle codes, we have shown that the performance of the iterative decoder is asymptotically as good as that of the ML decoder. As for tail-biting codes, the same thing is true if the lowest weight pseudocodeword is at least the minimum weight of the code. Unfortunately, the analysis of tail-biting codes and cycle codes does not extend to turbo codes and low density parity check codes in general. Our next approach is to determine the average behavior of message passing algorithms by studying the evolution of their "message" densities. For the class of "repeat and accumulate" serially concatenated turbo-like codes, we have devised an algorithm for determining their "threshold" values. When the signal-to-noise ratio is larger than the threshold value, the error probabilities of message passing algorithms approach zero, whereas if the signal-to-noise ratio is less than the threshold value, the error probability stays bounded away from zero. Some message passing algorithms and graphical representation of codes are efficient means of devising ML or MAP decoding algorithms. We have proposed a junction tree representation for linear block codes, and we have shown that the minimum junction tree can be less complex than the minimal trellis.