The field of error correcting codes was revolutionized by the introduction of turbo codes [7] in 1993. These codes demonstrated dramatic performance improvements over any previously known codes, with significantly lower complexity. Since then, much progress has been made towards understanding the performance of these codes, as well as in using this understanding to design even better codes.
This thesis takes a few more steps in both these directions. We develop a new technique, called the typical set bound, for analyzing the asymptotic performance of code ensembles based on their weight enumerators. This technique yields very tight bounds on the maximum-likelihood decoding threshold of code ensembles, and is powerful enough to reproduce Shannon's noisy coding theorem for the class of binary-input symmetric channels.
We also introduce a new class of codes called irregular repeat-accumulate~(IRA) codes, which are adapted from the previously known class of repeat-accumulate~(RA) codes. These codes are competitive in terms of decoding performance with the class of irregular low-density parity-check~(LDPC) codes, which are arguably the best class of codes known today, at least for long block lengths. In addition, IRA codes have a significant advantage over irregular LDPC codes in terms of encoding complexity.
We also derive an analytical bound regarding iterative decoding thresholds of code ensembles on general binary-input symmetric channels, an area in which theoretical results are currently lacking
The field of error correcting codes was revolutionized by the introduction of turbo codes [7] in 1993. These codes demonstrated dramatic performance improvements over any previously known codes, with significantly lower complexity. Since then, much progress has been made towards understanding the performance of these codes, as well as in using this understanding to design even better codes.
This thesis takes a few more steps in both these directions. We develop a new technique, called the typical set bound, for analyzing the asymptotic performance of code ensembles based on their weight enumerators. This technique yields very tight bounds on the maximum-likelihood decoding threshold of code ensembles, and is powerful enough to reproduce Shannon's noisy coding theorem for the class of binary-input symmetric channels.
We also introduce a new class of codes called irregular repeat-accumulate~(IRA) codes, which are adapted from the previously known class of repeat-accumulate~(RA) codes. These codes are competitive in terms of decoding performance with the class of irregular low-density parity-check~(LDPC) codes, which are arguably the best class of codes known today, at least for long block lengths. In addition, IRA codes have a significant advantage over irregular LDPC codes in terms of encoding complexity.
We also derive an analytical bound regarding iterative decoding thresholds of code ensembles on general binary-input symmetric channels, an area in which theoretical results are currently lacking
Low-density parity-check (LDPC) codes form a Shannon limit approaching class of linear block codes. With iterative decoding based on their Tanner graphs, they can achieve outstanding performance. Since their rediscovery in late 1990's, the design, construction, and decoding of LDPC codes as well as their generalization have become one of the focal research points. This thesis takes a few more steps in these directions. The first significant contribution of this thesis is the introduction of a new class of codes called Generalized Irregular Low-Density (GILD) parity-check codes, which are adapted from the previously known class of Generalized Low-Density (GLD) codes. GILD codes are generalization of irregular LDPC codes, and are shown to outperform GLD codes. In addition, GILD codes have a significant advantage over GLD codes in terms of encoding and decoding complexity. They are also able to match and even beat LDPC codes for small block lengths. The second significant contribution of this thesis is the proposition of several decoding algorithms. Two new decoding algolithms for LDPC codes are introduced. In principle and complexity these algorithms can be grouped with bit flipping algorithms. Two soft-input soft-output (SISO) decoding algorithms for linear block codes are also proposed. The first algorithm is based on Maximum a Posteriori Probability (MAP) decoding of low-weight subtrellis centered around a generated candidate codeword. The second algorithm modifies and utilizes the improved Kaneko's decoding algorithm for soft-input hard-output decoding. These hard outputs are converted to soft-decisions using reliability calculations. Simulation results indicate that the proposed algorithms provide a significant improvement in error performance over Chase-based algorithm and achieve practically optimal performance with a significant reduction in decoding complexity. An analytical expression for the union bound on the bit error probability of linear codes on the Gilbert-Elliott (GE) channel model is also derived. This analytical result is shown to be accurate in establishing the decoder performance in the range where obtaining sufficient data from simulation is impractical.
Channel coding is a key component of artificial communication systems, allowing reliable communication using unreliable channels. In the last decade, iteratively decoded channel codes have become or clearly will become standards in a wide range of applications where large amounts of information must be communicated using unreliable media. Of the class of iteratively decoded codes, Low-density parity check (LDPC) codes are arguably the simplest class to describe, and indeed were described more than four decades ago in 1963 by Robert Gallager. The current understanding of LDPC codes has progressed in several significant ways beyond what had been expressed by 1963 by Gallager. Importantly, irregular LDPC codes, whose parity check matrices do not have constant row and column sums, have been shown to significantly outperform their regular counterparts explicitly considered by Gallager. By 1999, researchers had defined a class of irregular ensembles, each characterized by a pair of polynomials. Along with this new class of ensemble, they defined an analytical technique, density evolution, that accurately predicted the channel coding performance of a typical code under iterative message-passing decoding. The pair of polynomials could be effectively be designed by optimizing the coefficients of the polynomial for density evolution threshold threshold. This thesis concerns a different class of ensembles, namely protograph ensembles. Protograph ensembles are characterized by a template graph called, intuitively, a protograph. The Tanner-graph representation of a code in the ensemble is a random lift of the protograph. Protograph-based codes have significant advantages over unstructured irregular codes in regard to implementation of their encoders and decoders. In the decoder, this structure can be used in at least two distinct ways to organize the computations defined by any message-passing algorithm. If, in addition to the protograph structure, circulant structure is imposed on each "section" of the matrix then a quasi-cyclic code results, bestowing even mores advantages, especially in the possible implementation of the encoder. A central difficulty in using protograph ensembles is finding a suitable protograph. Since graphs are discrete objects, there is no obvious correspondence to any optimization model using vectors of real numbers. Instead, the technique of simulated annealing has been applied with a remarkable degree of success. For example, on the AWGN channel, given a constraint on the node degrees, protograph ensembles can be found that achieve a threshold only half as far (measured in dB) from the Shannon limit as unstructured irregular ensembles. This simultaneously illustrates an inherent performance advantage of protograph codes over unstructured codes as well as the efficacy of simulated annealing as an optimization technique. A persistent problem which appears to be common in all codes optimized for density evolution threshold is that of error floors. On a superficial level, this is explained by the maxim that "There's no such thing as a free lunch." In some contexts, such as in codes designed for the erasure channel, the phenomenon can be explained on a much deeper level, though it is not clear why the phenomenon should persist so universally. Still, even without a detailed understanding the cause of this problem, there are techniques that can mitigate error floors. An important tool toward this end is weight enumerators, which are discussed in chapter 3. Codeword and stopping set enumerators can be efficiently computed if a certain (non-concave) function can be efficiently maximized. Protographs that are selected on the basis of their enumerators have shown some success in reducing error floors.
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.
We derive simple lower bounds on the number of iterations which are required to communicate over the binary erasure channel when graph-based code ensembles are used in conjunction with an iterative message-passing decoder. These bounds refer to the asymptotic case where we let the block length tend to infinity, and the bounds apply to general ensembles of low- density parity-check (LDPC) codes, irregular repeat-accumulate (IRA) and accumulate-repeat-accumulate (ARA) codes. It is demonstrated that, under a mild condition, the number of iterations required for successful decoding scales at least like the inverse of the gap (in rate) to capacity.
The previous chapter examined code properties responsible for the floor region (high-SNR region) of LDPC and turbo codes. Specifically, the emphasis was on the computation of various weight enumerators for LDPC and turbo code ensembles because a poor weight spectrum leads to poor performance in a code's floor region both for iterative decoders and for maximum-likelihood decoders. That chapter also introduced ensemble enumerators for trapping sets and stopping sets, both of which can lead to poor floor performance in iterative decoders. In this chapter we examine the iterative decoding performance of LDPC and turbo code ensembles in the complementary low-SNR region, the “waterfall” region. We show that the iterative decoding of long LDPC and turbo codes displays a threshold effect in which communication is reliable beyond this threshold and unreliable below it. The threshold is a function of code ensemble properties and the tools introduced in this chapter allow the designer to predict the decoding threshold and its gap from Shannon's limit. The ensemble properties for LDPC codes are the degree distributions which are typically the design targets for the code-design techniques presented in subsequent chapters. The ensemble properties for turbo codes are the selected constituent codes. Our development borrows heavily from the references listed at the end of the chapter and our focus is on the binary-input AWGN channel. The references and the problems consider other channels. Density Evolution for Regular LDPC Codes We first summarize the main results of [1] with regard to iterative (message passing) decoding of long, regular LDPC codes.
This thesis addresses the topic of Low-Density Parity-Check (LDPC) code analysis, both asymptotically and for finite block lengths. Since in general it is a difficult problem to analyze individual code instances, ensemble averages are studied by following Gallager's original idea and results of Luby et al. Often, one can relate the insights gained by studying ensemble averages to statements regarding individual codes by proving that most elements in the ensemble behave "close" to the average. One important such average is the average weight distribution, another is the average pseudo-codeword distribution, where pseudo-codewords play the same role under iterative decoding as codewords do for maximum likelihood decoding. One of the main contributions of this thesis is the calculation of such averages for fully irregular LDPC code ensembles. Much of classical coding theory is aimed at the construction of codes with large minimum distance. This is so since away from capacity accurate bounds on the performance of a code can be given in terms of its minimum distance (or more generally, in terms of its weight distribution). Therefore, it is of interest to investigate what role the minimum distance plays for iteratively decoded LDPC code ensembles. In particular, sequences of capacity-achieving LDPC code ensembles of increasing length when transmission takes place over the Binary Erasure Channel (BEC) are investigated. It is shown that, under certain technical conditions, the minimum distance of such ensembles grows sub-linearly in the block length, a result which is somewhat surprising from a classical point of view. Specific attention is also given to the design of LDPC code ensembles, where an inherent trade-off is observed between achieving large minimum distance (which is relevant for the so called error-floor regime) and achieving a large threshold (which in the limit of long block lengths determines the worst channel on which transmission can be accomplished reliably). Most results on iterative coding systems to date address their asymptotic performance, i.e., their performance when the block length tends to infinity. For small and moderate block lengths the behavior of a code can deviate significantly from its asymptotic limit. It is therefore of high practical value to be able to analyze the finite-length performance of LDPC code ensembles. In this thesis, an exact such analysis is presented for iteratively decoded LDPC code ensembles over the BEC. In particular, expressions for the exact average bit and block erasure probabilities are computed by solving a set of recursions. Such an analysis is the starting point for a finite-length optimization, a topic which is slated for future work. The methods used in this thesis include a combinatorial approach (familiar to the coding society) as well as the powerful techniques developed in the statistical physics community, for example, the replica method or the mean-field approximation. Although the statistical physics techniques are ideally suited for the analysis of iterative coding systems, they are to date only accessible to a relatively small community. Therefore, an overview of these techniques is first presented using a language familiar to the coding theory community. Next, in order to highlight the differences and common points between the combinatorial and the statistical physics approaches, both techniques are applied to the weight distribution problem. It turns out that for regular ensembles, both methods yield the same result, while for irregular ensembles, they do differ in general.