Low-density parity check (LDPC) codes have experienced tremendous popularity due to their capacity-achieving performance. In this dissertation, several different aspects of LDPC coding and its applications to magnetic storage are investigated. One of the most significant issues that impedes the use of LDPC codes in many systems is the error-rate floor phenomenon associated with their iterative decoders. By delineating the fundamental principles, we extend to partial response channels algorithms for predicting the error rate performance in the floor region for the binary-input AWGN channel. We develop three classes of decoding algorithms for mitigating the error floor by directly tackling the cause of the problem: trapping sets. In our experiments, these algorithms provide multiple orders of improvement over conventional decoders at the cost of various implementation complexity increases.Product codes are widely used in magnetic recording systems where errors are both isolated and bursty. A dual-mode decoding technique for Reed-Solomon-code-based product codes is proposed, where the second decoding mode involves maximum-likelihood erasure decoding of the binary images of the Reed-Solomon codewords. By exploring a tape storage application, we demonstrate that this dual-mode decoding system dramatically improves the performance of product codes. Moreover, the complexity added by the second decoding mode is manageable. We also show the performance of this technique on a product code which has an LDPC code in the columns.Run-length-limited (RLL) codes are ubiquitous in today's disk drives. Using RLL codes has enabled drive designers to pack data very efficiently onto the platter surface by ensuring stable symbol-timing recovery. We consider a concatenation system design with an LDPC code and an RLL code as components to simultaneously achieve desirable features such as: soft information availability to the LDPC decoder, the preservation of the LDPC code's structure, and the capability of correcting long erasure bursts.We analyze the performance of LDPC-coded magnetic recording channel in the presence of media noise. We employ advanced signal processing for the pattern-dependent-noise-predictive channel detectors, and demonstrate that a gain of over 1 dB or a linear density gain of about 8% relative to a comparable Reed-Solomon is attainable by using an LDPC code.

As one of the most powerful error-correcting codes, Low-density parity check codes are widely used in digital communications. Because of the performance of LDPC codes are capable to close the shannon limited extraordinarily, LDPC codes are to be used in the new Digital Video Broadcast-Satellite-Second Generation(DVB-S2) and it is the first time that LDPC codes are included in the broadcast standard in 2003. In this thesis, a restructured parity-check matrices which can be divided into sub-matrices for LDPC code in DVB-S2 is provided. Corresponded to this restructured parity-check matrix, a reconstructed decoding table is invented. The encoding table of DVB-S2 standard only could obtain the unknown check nodes from known variable nodes, while the decoding table this thesis provided could obtain the unknown variable nodes from known check nodes what is exactly the Layered-massage passing algorithm needed. Layered-message passing algorithm which also known as "Turbo-decoding message passing" is used to reduce the decoding iterations and memory storage for messages. The thesis also investigate Bp algorithm, lambda-min algorithm, Min-sum algorithm and SISO-s algorithm, meanwhile, simulation results of these algorithms and schedules are also presented.

We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular, the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m²,2, m] code as compared to the toric code which is a [2m²,2, m]code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

Στα πλαίσια αυτής της διπλωματικής εργασίας μελετήσαμε μεθόδους κωδικοποίησης με χρήση πινάκων ισοτιμίας μεγάλων διαστάσεων που έχουν χρησιμοποιηθεί και εφαρμοσθεί μέχρι τώρα σε προηγούμενες μελέτες. Επιλέξαμε τη σχεδίαση ενός αποκωδικοποιητή, που στηρίζεται στο WiMAX – 802.16e ΙΕΕΕ πρότυπο μετάδοσης και συγκεκριμένα με χρήση πομπού και δέκτη με περισσότερες από μία κεραίες. Παρουσιάζουμε, λοιπόν τη θεωρία που συσχετίζεται με το θέμα αυτό τόσο από την πλευρά της κωδικοποίησης όσο κι από την πλευρά της ασύρματης ΜΙΜΟ μετάδοσης και το πρότυπο WiMAX. Αναλύουμε κάθε τμήμα του συστήματός που προσομοιώνουμε και παραθέτουμε τα αποτελέσματα της προσομοίωσης. / -

The efficient and reliable communication of data from multiple sources to a single receiver plays an important role in emerging applications such as wireless sensor networks. The correlation among observations picked-up by spatially distributed sensors in such a network can be exploited to enhance the efficiency and reliability of communication. In particular, information theory shows that optimal communication of information from correlated sources requires distributed joint source-channel (DJSC) coding. This dissertation develops new approaches to designing DJSC codes based on low density parity check (LDPC) codes. The existence of low complexity code optimization algorithms and decoding algorithms make these codes ideal for joint optimization and decoding of multiple codes operating on correlated sources. The well known EXIT analysis-based LDPC code optimization method for channel coding in single-user point-to-point systems is extended to the optimization of two-user LDPC codes for DJSC coding in multi-access channels (MACs) with correlated users. Considering an orthogonal MAC with two correlated binary sources, an asymptotically optimal DJSC code construction capable of achieving any rate-pair in the theoretically-achievable two-user rate-region is presented. A practical approach to realizing this scheme using irregular LDPC codes is then developed. Experimental results are presented which demonstrate that the proposed codes can approach theoretical bounds when the codeword length is increased. For short codeword length and high inter-source correlation, these DJSC codes are shown to significantly outperform separate source and channel codes. Next, the DJSC code design for the transmission of a pair of correlated binary sources over a Gaussian MAC (GMAC) is investigated. The separate source and channel coding is known to be sub-optimal in this case. For the optimization of a pair of irregular LDPC codes, the EXIT analysis for message passing in a joint factor-graph decoder is analyzed, and an approach to modeling the probability density functions of messages associated with graph nodes which represent the inter-source dependence is proposed. Simulation results show that, for sufficiently large codeword lengths and high inter-source correlation, the proposed DJSC codes for GMAC can achieve rates higher than the theoretical upper bound for separate source and channel coding.

The efficient and reliable communication of data from multiple sources to a single receiver plays an important role in emerging applications such as wireless sensor networks. The correlation among observations picked-up by spatially distributed sensors in such a network can be exploited to enhance the efficiency and reliability of communication. In particular, information theory shows that optimal communication of information from correlated sources requires distributed joint source-channel (DJSC) coding. This dissertation develops new approaches to designing DJSC codes based on low density parity check (LDPC) codes. The existence of low complexity code optimization algorithms and decoding algorithms make these codes ideal for joint optimization and decoding of multiple codes operating on correlated sources. The well known EXIT analysis-based LDPC code optimization method for channel coding in single-user point-to-point systems is extended to the optimization of two-user LDPC codes for DJSC coding in multi-access channels (MACs) with correlated users. Considering an orthogonal MAC with two correlated binary sources, an asymptotically optimal DJSC code construction capable of achieving any rate-pair in the theoretically-achievable two-user rate-region is presented. A practical approach to realizing this scheme using irregular LDPC codes is then developed. Experimental results are presented which demonstrate that the proposed codes can approach theoretical bounds when the codeword length is increased. For short codeword length and high inter-source correlation, these DJSC codes are shown to significantly outperform separate source and channel codes. Next, the DJSC code design for the transmission of a pair of correlated binary sources over a Gaussian MAC (GMAC) is investigated. The separate source and channel coding is known to be sub-optimal in this case. For the optimization of a pair of irregular LDPC codes, the EXIT analysis for message passing in a joint factor-graph decoder is analyzed, and an approach to modeling the probability density functions of messages associated with graph nodes which represent the inter-source dependence is proposed. Simulation results show that, for sufficiently large codeword lengths and high inter-source correlation, the proposed DJSC codes for GMAC can achieve rates higher than the theoretical upper bound for separate source and channel coding.

In digital communication, information is sent as bits, which is corrupted by the noise present in wired/wireless medium known as the channel. The Low Density Parity Check (LDPC) codes are a family of error correction codes used in communication systems to detect and correct erroneous data at the receiver. Data is encoded with error correction coding at the transmitter and decoded at the receiver. The Noisy Gradient Descent BitFlip (NGDBF) decoding algorithm is a new algorithm with excellent decoding performance with relatively low implementation requirements. This dissertation aims to characterize the performance of the NGDBF algorithm. A simple improvement over NGDBF called the Re-decoded NGDBF (R-NGDBF) is proposed to enhance the performance of NGDBF decoding algorithm. A general method to estimate the decoding parameters of NGDBF is presented. The estimated parameters are then verified in a hardware implementation of the decoder to validate the accuracy of the estimation technique.

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L'objet de cette thèse est l'étude des codes LDPC quantiques. Dans un premier temps, nous travaillons sur des constructions topologiques de codes LDPC quantiques. Nous proposons de construire une famille de codes couleur basée sur des pavages hyperboliques. Nous étudions ensuite les paramètres d'une famille de codes basée sur des graphes de Cayley.Dans une seconde partie, nous examinons les performances de ces codes. Nous obtenons une borne supérieure sur les performances des codes LDPC quantiques réguliers sur le canal à effacement quantique. Ceci prouve que ces codes n'atteignent pas la capacité du canal à effacement quantique. Dans le cas du canal de dépolarisation, nous proposons un nouvel algorithme de décodage des codes couleur basé sur trois décodages de codes de surface. Nos simulations numériques montrent de bonnes performances dans le cas des codes couleur toriques.Pour finir, nous nous intéressons au phénomène de percolation. La question centrale de la théorie de la percolation est la détermination du seuil critique. Le calcul exacte de ce seuil est généralement difficile. Nous relions la probabilité de percolation dans certains pavages réguliers du plan hyperbolique à la probabilité d'erreur de décodage pour une famille de codes hyperboliques. Nous en déduisons une borne sur le seuil critique de ces pavages hyperboliques basée sur des résultats de théorie de l'information quantique. Il s'agit d'une application de la théorie de l'information quantique à un problème purement combinatoire. / This thesis is devoted to the study of quantum LDPC codes. The first part presents some topological constructions of quantum LDPC codes. We introduce a family of color codes based on tilings of the hyperbolic plane. We study the parameters of a family of codes based on Cayley graphs.In a second part, we analyze the performance of these codes. We obtain an upper bound on the performance of regular quantum LDPC codes over the quantum erasure channel. This implies that these codes don't achieve the capacity of the quantum erasure channel. In the case of the depolarizing channel, we propose a new decoding algorithm of color codes based on three surface codes decoding. Our numerical results show good performance for toric color codes.Finally, we focus on percolation theory. The central question in percolation theory is the determination of the critical probability. Computing the critical probability exactly is usually quite difficult. We relate the probability of percolation in some regular tilings of the hyperbolic plane to the probability of a decoding error for hyperbolic codes on the quantum erasure channel. This leads to an upper bound on the critical probability of these hyperbolic tilings based on quantum information. It is an application of quantum information to a purely combinatorial problem.

No description available.