How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published in IEEE Trans. Inform. Theor

Device-Independent Quantum Key Distribution

Cryptographic key exchange protocols traditionally rely on computational conjectures such as the hardness of prime factorisation to provide security against eavesdropping attacks. Remarkably, quantum key distribution protocols like the one proposed by Bennett and Brassard provide information-theoretic security against such attacks, a much stronger form of security unreachable by classical means. However, quantum protocols realised so far are subject to a new class of attacks exploiting implementation defects in the physical devices involved, as demonstrated in numerous ingenious experiments. Following the pioneering work of Ekert proposing the use of entanglement to bound an adversary's information from Bell's theorem, we present here the experimental realisation of a complete quantum key distribution protocol immune to these vulnerabilities. We achieve this by combining theoretical developments on finite-statistics analysis, error correction, and privacy amplification, with an event-ready scheme enabling the rapid generation of high-fidelity entanglement between two trapped-ion qubits connected by an optical fibre link. The secrecy of our key is guaranteed device-independently: it is based on the validity of quantum theory, and certified by measurement statistics observed during the experiment. Our result shows that provably secure cryptography with real-world devices is possible, and paves the way for further quantum information applications based on the device-independence principle.Comment: 5+1 pages in main text and methods with 4 figures and 1 table; 37 pages of supplementary materia

From Polar to Reed-Muller Codes:Unified Scaling, Non-standard Channels, and a Proven Conjecture

The year 2016, in which I am writing these words, marks the centenary of Claude Shannon, the father of information theory. In his landmark 1948 paper "A Mathematical Theory of Communication", Shannon established the largest rate at which reliable communication is possible, and he referred to it as the channel capacity. Since then, researchers have focused on the design of practical coding schemes that could approach such a limit. The road to channel capacity has been almost 70 years long and, after many ideas, occasional detours, and some rediscoveries, it has culminated in the description of low-complexity and provably capacity-achieving coding schemes, namely, polar codes and iterative codes based on sparse graphs. However, next-generation communication systems require an unprecedented performance improvement and the number of transmission settings relevant in applications is rapidly increasing. Hence, although Shannon's limit seems finally close at hand, new challenges are just around the corner. In this thesis, we trace a road that goes from polar to Reed-Muller codes and, by doing so, we investigate three main topics: unified scaling, non-standard channels, and capacity via symmetry. First, we consider unified scaling. A coding scheme is capacity-achieving when, for any rate smaller than capacity, the error probability tends to 0 as the block length becomes increasingly larger. However, the practitioner is often interested in more specific questions such as, "How much do we need to increase the block length in order to halve the gap between rate and capacity?". We focus our analysis on polar codes and develop a unified framework to rigorously analyze the scaling of the main parameters, i.e., block length, rate, error probability, and channel quality. Furthermore, in light of the recent success of a list decoding algorithm for polar codes, we provide scaling results on the performance of list decoders. Next, we deal with non-standard channels. When we say that a coding scheme achieves capacity, we typically consider binary memoryless symmetric channels. However, practical transmission scenarios often involve more complicated settings. For example, the downlink of a cellular system is modeled as a broadcast channel, and the communication on fiber links is inherently asymmetric. We propose provably optimal low-complexity solutions for these settings. In particular, we present a polar coding scheme that achieves the best known rate region for the broadcast channel, and we describe three paradigms to achieve the capacity of asymmetric channels. To do so, we develop general coding "primitives", such as the chaining construction that has already proved to be useful in a variety of communication problems. Finally, we show how to achieve capacity via symmetry. In the early days of coding theory, a popular paradigm consisted in exploiting the structure of algebraic codes to devise practical decoding algorithms. However, proving the optimality of such coding schemes remained an elusive goal. In particular, the conjecture that Reed-Muller codes achieve capacity dates back to the 1960s. We solve this open problem by showing that Reed-Muller codes and, in general, codes with sufficient symmetry are capacity-achieving over erasure channels under optimal MAP decoding. As the proof does not rely on the precise structure of the codes, we are able to show that symmetry alone guarantees optimal performance

Compression pour la communication interactive de contenus visuels

Interactive images and videos have received increasing attention due to the interesting features they provide. With these contents, users can navigate within the content and explore the scene from the viewpoint they desire. The characteristics of these media make their compression very challenging. On the one hand, the data is captured in high resolution (very large) to experience a real sense of immersion. On the other hand, the user requests a small portion of the content during navigation. This requires two characteristics: efficient compression of data by exploiting redundancies within the content (to lower the storage cost), and random access ability to extract part of the compressed stream requested by the user (to lower the transmission rate). Classical compression schemes can not handle random accessibility because they use a fixed pre-defined order of sources to capture redundancies.The purpose of this thesis is to provide new tools for interactive compression schemes of images. For that, as the first contribution, we propose an evaluation framework by which we can compare different image/video interactive compression schemes. Moreover, former theoretical studies show that random accessibility can be achieved using incremental codes with the same transmission cost as non-interactive schemes and with reasonable storage overhead. Our second contribution is to build a generic coding scheme that can deal with various interactive media. Using this generic coder, we then propose compression tools for 360-degree images and 3D model texture maps with random access ability to extract the requested part. We also propose new representations for these modalities. Finally, we study the effect of model selection on the compression rates of these interactive coders.Les images et vidéos interactives ont récemment vu croître leur popularité. En effet, avec ce type de contenu, les utilisateurs peuvent naviguer dans la scène et changer librement de point de vue. Les caractéristiques de ces supports posent de nouveaux défis pour la compression. D'une part, les données sont capturées en très haute résolution pour obtenir un réel sentiment d'immersion. D'autre part, seule une petite partie du contenu est visualisée par l'utilisateur lors de sa navigation. Cela induit deux caractéristiques : une compression efficace des données en exploitant les redondances au sein du contenu (pour réduire les coûts de stockage) et une compression avec accès aléatoire pour extraire la partie du flux compressé demandée par l'utilisateur (pour réduire le débit de transmission). Les schémas classiques de compression ne peuvent gérer de manière optimale l’accès aléatoire, car ils utilisent un ordre de traitement des données fixe et prédéfini qui ne peut s'adapter à la navigation de l'utilisateur.Le but de cette thèse est de fournir de nouveaux outils pour les schémas interactifs de compression d’images. Pour cela, comme première contribution, nous proposons un cadre d’évaluation permettant de comparer différents schémas interactifs de compression d'image / vidéo. En outre, des études théoriques antérieures ont montré que l’accès aléatoire peut être obtenu à l’aide de codes incrémentaux présentant le même coût de transmission que les schémas non interactifs au prix d'une faible augmentation du coût de stockage. Notre deuxième contribution consiste à créer un schéma de codage générique pouvant s'appliquer à divers supports interactifs. À l'aide de ce codeur générique, nous proposons ensuite des outils de compression pour deux modalités d'images interactives : les images omnidirectionnelles (360 degrés) et les cartes de texture de modèle 3D. Nous proposons également de nouvelles représentations de ces modalités. Enfin, nous étudions l’effet de la sélection du modèle sur les taux de compression de ces codeurs interactifs

Polar codes for data storage and communication network applications

The dissertation provides polar coding techniques for a variety of source and channel models with applications to storage and communication networks. We first provide universal polar codes for asymmetric compound channels that avoid common randomness. A staircase alignment of polar blocks is considered in the code construction. An MDS code is used in each column achieving the universality and a scrambling technique is implemented for each column helping avoid common randomness. These compound asymmetric channels are used for modelling flash-memories, such as MLCs (multi-level cell flash memories), and TLCs (three-level cell flash memories) memories. Hence the proposed universal polar codes for asymmetric channels can be used for flash memory error correction. The costly noiseless channel model was used to model a flash memory device. Each of the voltage levels to which a flash memory cell can be programmed has an associated wear cost which reflects the damage caused to the cell by repeated programming to that level. Shaping codes that minimize the average cost per channel symbol for a specified rate and shaping codes that minimize the average cost per source symbol (i.e., the total cost) have been shown to reduce cell wear and increase the lifetime of the memory. Hence, we study polar shaping codes for costly noiseless channels minimizing total cost. We also study polar shaping codes for costly noisy channels for the design of efficient codes that combine wear reduction and error correction for use in a noisy flash memory device. A novel scheme based on polar codes is proposed to compress a uniform source when a side information correlated with the source is available at the receiver while the conditional distribution of the side information given the source is symmetric and unknown to the source. An adaptation of universal polar codes with an incorporation of the linear code duality between channel coding and Slepian-Wolf coding is used in the design of those codes. Optimal rate is achieved through the proposed codes for the source model. These codes can be used in a wireless sensor network where the measurements tracked at two different nodes are correlated and the correlation may not always be fixed due to environmental changes such as weather. The nodes communicate the information sensed or measured by them to a central location. Finally, we provide a capacity-achieving polar coding strategy on a multi-level 3-receiver broadcast channel in which the second receiver is degraded (stochastically) from the first receiver for the transmission of a public message intended for all the receivers and a private message intended for the first receiver. A chaining strategy, translating the ideas of superposition coding, rate-splitting and indirect coding into polar coding, is used in the construction. The codes designed for such a channel model and setting can be used for video and audio file transfer in a client-server network where the individual clients are a computer and two mobile phones. The two mobile phone clients just support audio application where the computer supports both audio and video applications