A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multi-edge type LDPC codes.
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multi-edge type LDPC codes.
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
The design of optimal linear block codes capable of being efficiently decoded is of major concern, especially for short block lengths. As near capacity-approaching codes, Low-Density Parity-Check (LDPC) codes possess several advantages over other families of codes, the most notable being its efficient decoding via Belief Propagation. While many LDPC code design methods exist, the development of efficient sparse codes that meet the constraints of modern short code lengths and accommodate new channel models remains a challenge. In this work, we propose for the first time a gradient-based data-driven approach for the design of sparse codes. We develop locally optimal codes with respect to Belief Propagation decoding via the learning of the Factor graph under channel noise simulations. This is performed via a novel complete graph tensor representation of the Belief Propagation algorithm, optimized over finite fields via backpropagation and coupled with an efficient line-search method. The proposed approach is shown to outperform the decoding performance of existing popular codes by orders of magnitude and demonstrates the power of data-driven approaches for code design.
A LDPC code is binary linear code equipped with a sparse parity-check matrix. A bipartite graph, called Tanner graph, can be associated to a parity-check matrix. Then message passing iterative decoding (MPID) algorithms can be applied on the Tanner graph to do fast efficient decoding. Richardson and Urbanke proposed a scheme, called density evolution, to analyze the performance of MPID algorithms. Using density evolution we shall report new approaches to designing asymptotically good LDPC code ensembles over binary erasure channel (BEC). We then answer an open question raised by Shokrollahi and Oswald during their study on capacity-achieving sequences over BEC. Next, in order to understand MPID algorithms better we bring up the discussion of pseudo-codewords, which are regarded as the cause of decoding errors. We shall present two rational functions whose monomials in the Taylor expansion enumerate all the pseudo-codewords of a general binary parity-check code.
The best known practical algorithm for the decoding of low-density parity-check (LDPC) codes is the iterative sum-product or belief propagation algorithm, operating on a Tanner graph (TG) of the code. Conventionally, in each iteration, all the symbol nodes and subsequently all the check nodes in the TG pass new messages to their neighbors (the so-called "flooding schedule"). We propose a new message-passing schedule, called "probabilistic schedule". Unlike flooding, the probabilistic schedule operation is based on the structure of the TG, and probabilistically balances the frequency at which different symbol nodes update their outgoing messages in accordance with their girths. Our results show that, particularly for short block lengths, the new schedule offers a much better performance/complexity trade-off. For a particular example of a (1268, 456) code, at no cost in complexity, the probabilistic schedule not only decreases both the bit and message error rates by an order of magnitude, but also reduces the number of undetected errors considerably. This work shows the importance of scheduling in the performance of iterative decoding algorithms and suggests that a schedule which matches the structure of the TG can substantially improve the performance/complexity trade-off in the decoding of short LDPC codes.
We study a wide range of graph-based message-passing schedules for iterative decoding of low-density parity-check (LDPC) codes. Using the Tanner graph (TG) of the code and for different nodes and edges of the graph, we relate the first iteration in which the corresponding messages deviate from their optimal value (corresponding to a cycle-free graph) to the girths and the lengths of the shortest closed walks in the graph. Using this result, we propose schedules, which are designed based on the distribution of girths and closed walks in the TG of the code, and categorize them as node based versus edge based, unidirectional versus bidirectional, and deterministic versus probabilistic. These schedules, in some cases, outperform the previously known schedules, and in other cases, provide less complex alternatives with more or less the same performance. The performance/complexity tradeoff and the best choice of schedule appear to depend not only on the girth and closed-walk distributions of the TG, but also on the iterative decoding algorithm and channel characteristics. We examine the application of schedules to belief propagation (sum-product) over additive white Gaussian noise (AWGN) and Rayleigh fading channels, min-sum (max-sum) over an AWGN channel, and Gallager's algorithm A over a binary symmetric channel.