The Capacity of Online (Causal) qq-ary Error-Erasure Channels

In the $q$-ary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1,\ldots,q-1\}^n$ symbol by symbol via a channel limited to at most $pn$ errors and/or $p^{*} n$ erasures. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has a full knowledge on the sent codeword $\mathbf{x}$. In this work we study the capacity of $q$-ary online channels for a combined corruption model, in which the channel may impose at most $pn$ {\em errors} and at most $p^{*} n$ {\em erasures} on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we give a full characterization of the capacity as a function of $q,p$, and $p^{*}$.Comment: This is a new version of the binary case, which can be found at arXiv:1412.637

The benefit of a 1-bit jump-start, and the necessity of stochastic encoding, in jamming channels

We consider the problem of communicating a message $m$ in the presence of a malicious jamming adversary (Calvin), who can erase an arbitrary set of up to $pn$ bits, out of $n$ transmitted bits $(x_1,\ldots,x_n)$. The capacity of such a channel when Calvin is exactly causal, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends on his observations $(x_1,\ldots,x_i)$ was recently characterized to be $1-2p$. In this work we show two (perhaps) surprising phenomena. Firstly, we demonstrate via a novel code construction that if Calvin is delayed by even a single bit, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends only on $(x_1,\ldots,x_{i-1})$ (and is independent of the "current bit" $x_i$) then the capacity increases to $1-p$ when the encoder is allowed to be stochastic. Secondly, we show via a novel jamming strategy for Calvin that, in the single-bit-delay setting, if the encoding is deterministic (i.e. the transmitted codeword is a deterministic function of the message $m$) then no rate asymptotically larger than $1-2p$ is possible with vanishing probability of error, hence stochastic encoding (using private randomness at the encoder) is essential to achieve the capacity of $1-p$ against a one-bit-delayed Calvin.Comment: 21 pages, 4 figures, extended draft of submission to ISIT 201

A characterization of the capacity of online (causal) binary channels

In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1\}^n$ bit by bit via a channel limited to at most $pn$ corruptions. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th bit or not based on its view so far, i.e., its decision depends only on the transmitted bits $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has a full knowledge on the sent codeword $\mathbf{x}$. In this work we study the capacity of binary online channels for two corruption models: the {\em bit-flip} model in which the channel may flip at most $pn$ of the bits of the transmitted codeword, and the {\em erasure} model in which the channel may erase at most $pn$ bits of the transmitted codeword. Specifically, for both error models we give a full characterization of the capacity as a function of $p$. The online channel (in both the bit-flip and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we present and analyze a coding scheme that improves on the previously suggested lower bounds and matches the previously suggested upper bounds thus implying a tight characterization

Generalized List Decoding

This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip ($\textsf{XOR}$) channels, erasure channels, $\textsf{AND}$ ($Z$-) channels, $\textsf{OR}$ channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) $(L-1)$-list decodable codes (where the list size $L$ is a universal constant) exist for such channels. Our criterion asserts that: "For any given general adversarial channel, it is possible to construct positive rate $(L-1)$-list decodable codes if and only if the set of completely positive tensors of order-$L$ with admissible marginals is not entirely contained in the order-$L$ confusability set associated to the channel." The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 1. extracting equicoupled subcodes (generalization of equidistant code) from any large code sequence using hypergraph Ramsey's theorem, and 2. significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which be may of independent interest. Other results include 1. List decoding capacity with asymptotically large $L$ for general adversarial channels; 2. A tight list size bound for most constant composition codes (generalization of constant weight codes); 3. Rederivation and demystification of Blinovsky's [Bli86] characterization of the list decoding Plotkin points (threshold at which large codes are impossible); 4. Evaluation of general bounds ([WBBJ]) for unique decoding in the error correction code setting

Lecture Notes on Network Information Theory

These lecture notes have been converted to a book titled Network Information Theory published recently by Cambridge University Press. This book provides a significantly expanded exposition of the material in the lecture notes as well as problems and bibliographic notes at the end of each chapter. The authors are currently preparing a set of slides based on the book that will be posted in the second half of 2012. More information about the book can be found at http://www.cambridge.org/9781107008731/. The previous (and obsolete) version of the lecture notes can be found at http://arxiv.org/abs/1001.3404v4/

Asymmetric Error Correction and Flash-Memory Rewriting using Polar Codes

We propose efficient coding schemes for two communication settings: 1. asymmetric channels, and 2. channels with an informed encoder. These settings are important in non-volatile memories, as well as optical and broadcast communication. The schemes are based on non-linear polar codes, and they build on and improve recent work on these settings. In asymmetric channels, we tackle the exponential storage requirement of previously known schemes, that resulted from the use of large Boolean functions. We propose an improved scheme, that achieves the capacity of asymmetric channels with polynomial computational complexity and storage requirement. The proposed non-linear scheme is then generalized to the setting of channel coding with an informed encoder, using a multicoding technique. We consider specific instances of the scheme for flash memories, that incorporate error-correction capabilities together with rewriting. Since the considered codes are non-linear, they eliminate the requirement of previously known schemes (called polar write-once-memory codes) for shared randomness between the encoder and the decoder. Finally, we mention that the multicoding scheme is also useful for broadcast communication in Marton's region, improving upon previous schemes for this setting.Comment: Submitted to IEEE Transactions on Information Theory. Partially presented at ISIT 201

Database Matching Under Noisy Synchronization Errors

The re-identification or de-anonymization of users from anonymized data through matching with publicly available correlated user data has raised privacy concerns, leading to the complementary measure of obfuscation in addition to anonymization. Recent research provides a fundamental understanding of the conditions under which privacy attacks, in the form of database matching, are successful in the presence of obfuscation. Motivated by synchronization errors stemming from the sampling of time-indexed databases, this paper presents a unified framework considering both obfuscation and synchronization errors and investigates the matching of databases under noisy entry repetitions. By investigating different structures for the repetition pattern, replica detection and seeded deletion detection algorithms are devised and sufficient and necessary conditions for successful matching are derived. Finally, the impacts of some variations of the underlying assumptions, such as the adversarial deletion model, seedless database matching, and zero-rate regime, on the results are discussed. Overall, our results provide insights into the privacy-preserving publication of anonymized and obfuscated time-indexed data as well as the closely related problem of the capacity of synchronization channels

Une odyssée de la communication classique au calcul quantique tolérant aux fautes

Cette thèse traite principalement de la protection de l'information. Non pas au sens de protection des renseignements privés dont on entend souvent parler dans les médias, mais plutôt au sens de robustesse à la corruption des données. En effet, lorsque nous utilisons un cellulaire pour envoyer un texto, plusieurs facteurs, comme les particules atmosphériques et l'interférence avec d'autres signaux, peuvent modifier le message initial. Si nous ne faisons rien pour protéger le signal, il est peu probable que le contenu du texto reste inchangé lors de la réception. C'est ce problème qui a motivé le premier projet de recherche de cette thèse. Sous la supervision du professeur David Poulin, j'ai étudié une généralisation des codes polaires, une technologie au coeur du protocole de télécommunication de 5\textsuperscript{ième} génération (5G). Pour cela, j'ai utilisé les réseaux de tenseurs, outils mathématiques initialement développés pour étudier les matériaux quantiques. L'avantage de cette approche est qu'elle permet une représentation graphique intuitive du problème, ce qui facilite grandement le développement des algorithmes. À la suite de cela, j'ai étudié l'impact de deux paramètres clés sur la performance des codes polaires convolutifs. En considérant le temps d'exécution des protocoles, j'ai identifié les valeurs de paramètres qui permettent de mieux protéger l'information à un coût raisonnable. Ce résultat permet ainsi de mieux comprendre comment améliorer les performances des codes polaires, ce qui a un grand potentiel d'application en raison de l'importance de ces derniers. Cette idée d'utiliser des outils mathématiques graphiques pour étudier des problèmes de protection de l'information sera le fil conducteur dans le reste de la thèse. Cependant, pour la suite, les erreurs n'affecteront plus des systèmes de communications classiques, mais plutôt des systèmes de calcul quantique. Comme je le présenterai dans cette thèse, les systèmes quantiques sont naturellement beaucoup plus sensibles aux erreurs. À cet égard, j'ai effectué un stage au sein de l'équipe de Microsoft Research, principalement sous la supervision de Michael Beverland, au cours duquel j'ai conçu des circuits permettant de mesurer un système quantique afin d'identifier les potentielles fautes qui affectent celui-ci. Avec le reste de l'équipe, nous avons prouvé mathématiquement que les circuits que j'ai développés sont optimaux. Ensuite, j'ai proposé une architecture pour implémenter ces circuits de façon plus réaliste en laboratoire et les simulations numériques que j'ai effectuées ont démontré des résultats prometteurs pour cette approche. D'ailleurs, ce résultat a été accueilli avec grand intérêt par la communauté scientifique et a été publié dans la prestigieuse revue \textit{Physical Review Letters}. Pour complémenter ce travail, j'ai collaboré avec l'équipe de Microsoft pour démontrer analytiquement que les architectures actuelles d'ordinateurs quantiques reposant sur des connexions locales entre les qubits ne suffiront pas pour la réalisation d'ordinateurs de grandes tailles protégés des erreurs. L'ensemble de ces résultats sont inspirés de méthodes issues de la théorie des graphes et plus particulièrement des méthodes de représentation des graphes dans un espace en deux dimensions. L'utilisation de telles méthodes pour le design de circuits et d'architectures quantiques est également une approche novatrice. J'ai terminé ma thèse sous la supervision du professeur Stefanos Kourtis. Avec celui-ci, j'ai créé une méthode, fondée sur la théorie des graphes et des méthodes d'informatique théorique, qui permet de concevoir automatiquement de nouveaux protocoles de correction des erreurs dans un système quantique. La méthode que j'ai conçue repose sur la résolution d'un problème de satisfaction de contraintes. Ce type de problème est généralement très difficile à résoudre. Cependant, il existe pour ces derniers un paramètre critique. En variant ce paramètre, le système passe d'une phase où les instances sont facilement résolubles vers une phase où il est facile de montrer qu'il n'y pas de solution. Les problèmes difficiles sont alors concentrés autour de cette transition. À l'aide d'expériences numériques, j'ai montré que la méthode proposée a un comportement similaire. Cela permet de montrer qu'il existe un régime où il est beaucoup plus facile que ce que le croyait la communauté de concevoir des protocoles de corrections des erreurs quantiques. De plus, en autant que je sache,l'article qui a résulté de ce travail est le premier qui met de l'avant ce lien entre la construction de protocoles de corrections des erreurs, les problèmes de satisfaction de contraintes et les transitions de phases