Asymptotic and finite-length optimization of LDPC codes

This thesis addresses the problem of information transmission over noisy channels. In 1993, Berrou, Glavieux and Thitimajshima discovered Turbo-codes. These codes made it possible to achieve a very good performance at low decoding complexity. In hindsight, it was recognized that the underlying principle of using sparse graph codes in conjunction with message-passing decoding was the same as the one proposed in Gallager's remarkable thesis of 1963, although Gallager's work had all but been forgotten in the intervening 30 years. In 1997, there occurred a breakthrough in the analysis of this type of codes. Luby, Mitzenmacher, Shokrollahi, Spielman and Stemann were able to give a complete characterization of the behavior of Low-Density Parity-Check code ensembles in the infinite blocklength case when used over the binary erasure channel. Soon thereafter, Richardson and Urbanke extended their results to binary-input memoryless symmetric channels. In this thesis we present tools to analyze the performance of these types of codes and ways to optimize their parameters. The optimization for the infinite blocklength case is relatively straightforward and we give a simple and efficient method of doing so. However, this does not really solve the problem in practice, since the asymptotic analysis has only limited relevance for the short or moderate blocklengths that are typically used in practice. This brings us to the main objective of this thesis, which is to bridge the gap between the asymptotic case and the practical finite-length case. We follow the lead of Luby et al. by considering the binary erasure channel. We show that the performance of LDPC codes obeys a well-defined scaling law as the blocklength increases. This scaling law refines the asymptotic analysis and provides a good way to understand and approximate the behavior of LDPC codes of short to moderate length. We show how to compute the scaling parameters involved and demonstrate how to use the resulting approximation as a design and optimization tool.

Coding for Communications and Secrecy

Mani Bastaniparizi

Shannon, in his landmark 1948 paper, developed a framework for characterizing the fundamental limits of information transmission. Among other results, he showed that reliable communication over a channel is possible at any rate below its capacity. In 2008, Arikan discovered polar codes; the only class of explicitly constructed low-complexity codes that achieve the capacity of any binary-input memoryless symmetric-output channel. Arikan's polar transform turns independent copies of a noisy channel into a collection of synthetic almost-noiseless and almost-useless channels. Polar codes are realized by sending data bits over the almost-noiseless channels and recovering them by using a low-complexity successive-cancellation (SC) decoder, at the receiver. In the first part of this thesis, we study polar codes for communications. When the underlying channel is an erasure channel, we show that almost all correlation coefficients between the erasure events of the synthetic channels decay rapidly. Hence, the sum of the erasure probabilities of the information-carrying channels is a tight estimate of the block-error probability of polar codes when used for communication over the erasure channel. We study SC list (SCL) decoding, a method for boosting the performance of short polar codes. We prove that the method has a numerically stable formulation in log-likelihood ratios. In hardware, this formulation increases the decoding throughput by 53% and reduces the decoder's size about 33%. We present empirical results on the trade-off between the length of the CRC and the performance gains in a CRC-aided version of the list decoder. We also make numerical comparisons of the performance of long polar codes under SC decoding with that of short polar codes under SCL decoding. Shannon's framework also quantifies the secrecy of communications. Wyner, in 1975, proposed a model for communications in the presence of an eavesdropper. It was shown that, at rates below the secrecy capacity, there exist reliable communication schemes in which the amount of information leaked to the eavesdropper decays exponentially in the block-length of the code. In the second part of this thesis, we study the rate of this decay. We derive the exact exponential decay rate of the ensemble-average of the information leaked to the eavesdropper in Wyner's model when a randomly constructed code is used for secure communications. For codes sampled from the ensemble of i.i.d. random codes, we show that the previously known lower bound to the exponent is exact. Our ensemble-optimal exponent for random constant-composition codes improves the lower bound extant in the literature. Finally, we show that random linear codes have the same secrecy power as i.i.d. random codes. The key to securing messages against an eavesdropper is to exploit the randomness of her communication channel so that the statistics of her observation resembles that of a pure noise process for any sent message. We study the effect of feedback on this approximation and show that it does not reduce the minimum entropy rate required to approximate a given process. However, we give examples where variable-length schemes achieve much larger exponents in this approximation in the presence of feedback than the exponents in systems without feedback. Upper-bounding the best exponent that block codes attain, we conclude that variable-length coding is necessary for achieving the improved exponents.

EPFL2017

Linear Universal Decoder for Compound Discrete Memoryless Channels

Rethnakaran Pulikkoonattu

Shannon in his seminal work \cite{paper:shannon} formalized the framework on the problem of digital communication of information and storage. He quantified the fundamental limits of compression and transmission rates. The quantity \textit{channel capacity} was coined to give an operational meaning on the ultimate limit of information transfer between two entities, transmitter and receiver. By using a random coding argument, he showed the achievability of reliable communication on any noisy channel as long as the rate is less than the channel capacity. More precisely, he argued that, using a fixed composition random codebook at the sender and a maximum likelihood decoder in the receiver, one can achieve almost zero error communication of information. Shannon's result was a promise on the achievability on the amount of data one can transfer over a medium. Achieving those promises has spurred scientific interest since 1948. Significant effort has been spent to find practical codes which gets close to the Shannon limits. For some class of channels, codes which are capacity achieving has been found since then. Low Density Parity Check Codes (LDPC) is one such capacity achieving family of codes for the class of binary erasure channels. Recently, Arikan proposed a family of codes known as Polar codes which are found to be capacity achieving for binary input memoryless channels. Slowly but steadily the longstanding problem of achieving Shannon limit is thus getting settled. In order to realize a practical system which can achieve the Shannon limits, two important things are necessary. One of them is an efficient capacity achieving code sequence and the other is an implementable decoding rule. Both these require the knowledge of the probabilistic law of the channel through which communication take place. At the transmitter one needs to know the channel capacity whereas at the receiver a perfect knowledge of the channel is necessitated to build optimum decoder. The question of whether a decoder can be designed without knowing the underlying channel, yet we can do a reliable communication arises here. This is the subject of discussion in this report. Consider a situation where a communication system is to be designed without explicit knowledge about the channel. Here, neither the transmitter nor the receiver knows the exact channel law. Suppose, the possible list of channels is made available upfront to both entities (transmitter and receiver). Can we devise a reliable communication scheme, irrespective of the actual channel picked without both transmitter and receiver being aware of it. This is the central theme in what is known\footnote{Also referred as \textit{coding for class of channels} as discussed in \cite{paper:blackwell}, \cite{book:Wolfowitz}.} as \textit{coding for compound channels}\cite{book:Csiszar}. Designing a reliable communication system for compound channels essentially involve building a codebook at the transmitter and a decoding rule at the receiver. The encoder-decoder pair together must guarantee reliable communication over a set of channels. In the general setting, no feedback is assumed from the receiver to the transmitter. Thus, a single coding strategy (codebook) must be devised (and fixed) upfront prior to transmission. At the receiver end, the decoding strategy must be devised independent of any knowledge of the channel. Of course, the codebook devised at the sender is perfectly known to the receiver. The answer to the question on whether such a communication system can be build is on the affirmative. In order to talk about the existence of reliable communication strategies, one must first talk about the maximum possible rate, the capacity. The highest achievable rate in a compound setting is known as the \textit{compound capacity} or capacity of the family. This is analogous to the famous Shannon capacity being the ultimate limit on achievable rate for any single channel. The capacity of compound set of memoryless channels has been studied by Blackwell et al in \cite{paper:blackwell} and that of linear dispersive channels is investigated by Root and Varaiya in \cite{paper:root:varaiya}. Lapidoth and Telatar looked at the compound capacity for families of channels with memory, more specifically the finite state channels \cite{paper:lapidoth-telatar1998}. As a special case, they have also derived the compound channel capacity of a class of Gilbert-Elliot channels. A decoder which operates without the knowledge of the channel in this setup is called a universal decoder. It is known that Maximum Mutual Information (MMI) decoders proposed by Goppa are universal. MMI decoders compute the empirical mutual between a received codeword against the codebook and find the best matching word as the true estimate. The complexity of MMI decoder remain fixed even if we were to find structured codes. This motivate us to ask the question whether we can build a universal decoder which offer better structure. Decoding rules which brings additive nature were considered in the literature as a potential scheme. Our work in this has been driven by this line of thoughts. In this work, we focus on class of discrete memoryless channels (DMC) and more specifically binary memoryless channels. We show that it is possible to build a linear universal decoder for any compound binary memoryless channels. The recent introduction of Polar codes motivated us to look into their suitability to the compound channel setup. We have carried out a preliminary investigation in this direction. While it is not clear whether Polar codes are universal under the optimum universal decoding, we find that they are universal for certain restricted classes of compound BMC sets.

2009

Asymptotic and finite-length analysis of low-density parity-check codes

Changyan Di

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.

EPFL2004