Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

Novel LDPC coding and decoding strategies: design, analysis, and algorithms

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

Extended Non-Binary Low-Density Parity-Check Codes over Erasure Channels

Based on the extended binary image of non-binary LDPC codes, we propose a method for generating extra redundant bits, such as to decreases the coding rate of a mother code. The proposed method allows for using the same decoder, regardless of how many extra redundant bits have been produced, which considerably increases the flexibility of the system without significantly increasing its complexity. Extended codes are also optimized for the binary erasure channel, by using density evolution methods. Nevertheless, the results presented in this paper can easily be extrapolated to more general channel models.Comment: ISIT 2011, submitte

LDPC-coded modulation for transmission over AWGN and flat rayleigh fading channels

La modulation codée est une technique de transmission efficace en largeur de bande qui intègre le codage de canal et la modulation en une seule entité et ce, afin d'améliorer les performances tout en conservant la même efficacité spectrale comparé à la modulation non codée. Les codes de parité à faible densité (low-density parity-check codes, LDPC) sont les codes correcteurs d'erreurs les plus puissants et approchent la limite de Shannon, tout en ayant une complexité de décodage relativement faible. L'idée de combiner les codes LDPC et la modulation efficace en largeur de bande a donc été considérée par de nombreux chercheurs. Dans ce mémoire, nous étudions une méthode de modulation codée à la fois puissante et efficace en largeur de bande, ayant d'excellentes performances de taux d'erreur binaire et une complexité d'implantation faible. Ceci est réalisé en utilisant un encodeur rapide, un décoder de faible complexité et aucun entrelaceur. Les performances du système proposé pour des transmissions sur un canal additif gaussien blanc et un canal à évanouissements plats de Rayleigh sont évaluées au moyen de simulations. Les résultats numériques montrent que la méthode de modulation codée utilisant la modulation d'amplitude en quadrature à M niveaux (M-QAM) peut atteindre d'excellentes performances pour toute une gamme d'efficacité spectrale. Une autre contribution de ce mémoire est une méthode simple pour réaliser une modulation codée adaptative avec les codes LDPC pour la transmission sur des canaux à évanouissements plats et lents de Rayleigh. Dans cette méthode, six combinaisons de paires encodeur modulateur sont employées pour une adaptation trame par trame. L'efficacité spectrale moyenne varie entre 0.5 et 5 bits/s/Hz lors de la transmission. Les résultats de simulation montrent que la modulation codée adaptative avec les codes LDPC offre une meilleure efficacité spectrale tout en maintenant une performance d'erreur acceptable

Improved linear programming decoding of LDPC codes and bounds on the minimum and fractional distance

We examine LDPC codes decoded using linear programming (LP). Four contributions to the LP framework are presented. First, a new method of tightening the LP relaxation, and thus improving the LP decoder, is proposed. Second, we present an algorithm which calculates a lower bound on the minimum distance of a specific code. This algorithm exhibits complexity which scales quadratically with the block length. Third, we propose a method to obtain a tight lower bound on the fractional distance, also with quadratic complexity, and thus less than previously-existing methods. Finally, we show how the fundamental LP polytope for generalized LDPC codes and nonbinary LDPC codes can be obtained.Comment: 17 pages, 8 figures, Submitted to IEEE Transactions on Information Theor