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

Abstract

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.