Polar codes were invented as the first error-correcting codes to achieve the capacity for the discrete channels with relatively low-complexity for encoding and decoding. However, this is only possible with significantly large code lengths which are not practical for many systems. Meanwhile, the performance of the finite-lengths polar codes is not competitive with other modern error-correcting codes. This can be attributed to the suboptimality of the decoding process and the relatively poor minimum Hamming distances. This thesis aims to improve the performance of polar codes. The contributions include improving the performance of the conventional successive cancellation decoder. This is based on a novel technique, namely one-step decision delay, which incorporates some extra computational nodes to the code tree. Also, this thesis presents two methods for increasing the Hamming distances of polar codes; in the first, the code rate remains unchanged, while in the second, that produces superior performance, a modest reduction in the code rate occurs. Both methods enhance the performance with using belief propagation decoder. In addition, the latency of the decoding process is reduced by applying the fast Hadamard transform decoder on polar codes for the first time. In this regard, a method to modify the encoder is presented for some lengths that are not normally compatible with the proposed decoder. Interestingly, this modification method has another advantage that in the sense that it increases the minimum Hamming distances of the codes. Furthermore, this thesis presents the utilisation of polar codes in some practical communication systems. Firstly, polar codes are examined over power line communication systems. In this respect, the construction of polar codes is analysed by using three different methods. Additionally, the practical performance of polar codes is illustrated for both single-carrier and multi-carrier channels. Secondly, this thesis explores polar codes for unequal error protection by partitioning the information sets into a number of subsets with different reliabilities. JPEG2000, which is an image compression method, is used for validating the advantages of the proposed technique. Moreover, a new joint source channel decoding is proposed from the combination of the polar decoder and JPEG2000 decoder in order to enhance the quality of the compressed images. Finally, polar codes are investigated for the energy transfer by presenting new subcodes that are chosen according to the weights of the produced codewords. In this context, two modes of energy transfer are presented. The proposed modes can successfully extend the lifetime of the receiver's battery.

Powerful error-correcting codes have enabled a dramatic increase in the bit density on the recording medium of hard-disk drives (HDDs). Error-correcting codes in magnetic recording require a low-complexity decoder and a code design that delivers a target error-rate performance. This dissertation proposes an error-correcting system based on polar codes incorporating a fast, low-complexity, soft-output decoder and a design that is optimized for error-rate performance in the magnetic recording channel. LDPC codes are the state-of-the-art in HDDs, providing the required error-rate performance on high densities at the cost of increased computational complexity of the decoder. Substantial research in LDPC codes has focused on reducing decoder complexity and has resulted in many variants such as quasi-cyclic and convolutional LDPC codes. Polar codes are a recent and important breakthrough in coding theory, as they achieve capacity on a wide spectrum of channels using a low-complexity successive cancellation decoder. Polar codes make a strong case for magnetic recording, because they have low complexity decoders and adequate finite-length error-rate performance. In their current form, polar codes are not feasible for magnetic recording for two reasons. Firstly, there is no low-complexity soft-output decoder available for polar codes that is required for turbo-based equalization of the magnetic recording channel. The only soft-output decoder available to date is a message passing based belief propagation decoder that has very high computational complexity and is not suitable for practical implementations. Secondly, current polar codes are optimized for the AWGN channel only, and may not perform well under turbo-based detector for ISI channels. This thesis delivers a powerful low-complexity error-correcting system based on polar codes for ISI channels. Specifically, we propose a low-complexity soft-output decoder for polar codes that achieves better error-rate performance than the belief propagation decoder for polar codes while drastically reducing the complexity. We further propose a technique for polar code design over ISI channels that outperform codes for the AWGN channel in terms of error rate under the proposed soft-output decoder.

In today’s society the ability to communicate with one another has grown, were a lot of focus is aimed towards speed in the telecommunication industry. For transmissions to become even faster, there are many ways to enhance transmission speeds of which error correction is one. Padding messages such that they are protected from noise, while using as few bits as possible and ensuring safe transmit is handled by error correction codes. Short codes with low complexity is a solution to faster transmission speeds. An error correction code which has gained a lot of attention since its first appearance in 2009 is Polar Codes. Polar Codes was chosen as the 3GPP standard for 5G control channel. The goal of the thesis is to develop and implement Polar Codes and rate matching according to the 3GPP standard 38.212. Polar Codes are then to be evaluated with different block sizes and rate matching settings. Finally Polar Code is compared with Convolutional code in a LTE-simulation environment. The performance evaluations are presented using BLER/(Eb/N0)-graphs. In this thesis a Polar encoder, rate matching and a Polar decoder (with Successive Cancellation algorithm) were successfully implemented. The simulation results show that Polar Codes performs better with longer block sizes and also has a better BLER-performance than Convolutional Codes when given the same message lengths.

Les réseaux de 5ème génération se caractérisent par la communication directe entre machines (M2M) et l’Internet des Objets, un réseau unifié d’objets connectés. Dans ce contexte, les appareils communicants sont des décideurs autonomes qui coopérent, coordonnent leurs actions et se reconfigurent de manière dynamique enfonction de leur environnement. L’enjeu est de développer des algorithmes efficaces pour coordonner les actions des appareils autonomes constituant le réseau.La théorie de l’information nous permet d’étudier le comportement de long-terme des appareils grâce aux distributions de probabilité conjointes. En particulier, nous sommes intéressés par la coordination forte, qui exige que la distribution induite sur les suites d’actions converge en distance L^1 vers une distribution i.i.d. cible.Nous considérons un model point-à-point composé d’une source d’information, d’un encodeur, d’un canal bruité, d’un décodeur, d’une information commune et nous cherchons à coordonner les signaux en entrée et en sortie du canal avec la source et sa reconstruction.Nos premiers résultats sont des bornes intérieures et extérieure pour la région de coordination forte, c’est-à-dire l’ensemble des distributions de probabilité conjointes réalisables et la quantité d’information commune requise.Ensuite, nous caractérisons cette région de coordination forte dans trois cas particuliers: lorsque le canal est parfait, lorsque le décodeur est sans perte et lorsque les variables aléatoires du canal sont indépendantes des variables aléatoires de la source. L’étude de ce dernier cas nous permet de remettre en cause le principe de séparation source-canal pour la coordination forte. Nous démontrons également que la coordination forte offre “gratuitement” des garanties de sécurité au niveau de la couche physique.Par ailleurs, nous étudions la coordination sous l’angle du codage polaire afin de développer des algorithmes de codage implémentables. Nous appliquons la polarisation de la source de manière à créer un schéma de codage explicite qui offre une alternative constructive aux preuves de codage aléatoires. / 5G networks will be characterized by machine to machine communication and the Internet of Things, a unified network of connected objects. In this context, communicating devices are autonomous decision-makers that cooperate, coordinate their actions, and reconfigure dynamically according to changes in the environment.To do this, it is essential to develop effective techniques for coordinating the actions of the nodes in the network.Information theory allows us to study the long-term behavior of the devices through the analysis of the joint probability distribution of their actions. In particular, we are interested in strong coordination, which requires the joint distribution of sequences of actions to converge to an i.i.d. target distribution in L^1 distance.We consider a two-node network comprised of an information source and a noisy channel, and we require the coordination of the signals at the input and at the output of the channel with the source and the reconstruction. We assume that the encoder and decoder share a common source of randomness and we introduce a state capturing theeffect of the environment.The first objective of this work is to characterize the strong coordination region, i.e. the set of achievable joint behaviors and the required minimal rates of common randomness. We prove inner and outer bounds for this region. Then, we characterize the exact coordination region in three particular cases: when the channel is perfect, when the decoder is lossless and when the random variables of the channel are separated from the random variables of the source.The study of the latter case allows us to show that the joint source-channel separation principle does not hold for strong coordination. Moreover, we prove that strong coordination offers “free” security guarantees at the physical layer.The second objective of this work is to develop practical codes for coordination: by exploiting the technique of source polarization, we design an explicit coding scheme for coordination, providing a constructive alternative to random coding proofs.

The objective of the proposed research is to develop and analyze coding schemes for information-theoretic security, which could bridge a gap between theory an practice. We focus on two fundamental models for information-theoretic security: secret-key generation for a source model and secure communication over the wire-tap channel. Many results for these models only provide existence of codes, and few attempts have been made to design practical schemes. The schemes we would like to propose should account for practical constraints. Specifically, we formulate the following constraints to avoid oversimplifying the problems. We should assume: (1) computationally bounded legitimate users and not solely rely on proofs showing existence of code with exponential complexity in the block-length; (2) a rate-limited public communication channel for the secret-key generation model, to account for bandwidth constraints; (3) a non-uniform and rate-limited source of randomness at the encoder for the wire-tap channel model, since a perfectly uniform and rate-unlimited source of randomness might be an expensive resource. Our work focuses on developing schemes for secret-key generation and the wire-tap channel that satisfy subsets of the aforementioned constraints.

Polar codes, recently invented by Arikan, are the first class of codes known to achieve the symmetric capacity for a large class of channels. The symmetric capacity is the highest achievable rate subject to using the binary input letters of the channel with equal probability. Polar code construction is based on a phenomenon called channel polarization. The encoding as well as the decoding operation of polar codes can be implemented with O(N logN) complexity, where N is the blocklength of the code. In this work, we study the factor graph representation of finite-length polar codes and their effect on the belief propagation (BP) decoding process over Binary Erasure Channel (BEC). Particularly, we study the parity-check-based (H-Based) as well as the generator based (G-based) factor graphs of polar codes. As these factor graphs are not unique for a code, we study and compare the performance of Belief Propagation (BP) decoders on number of well-known graphs. Error rates and complexities are reported for a number of cases. Comparisons are also made with the Successive Cancellation (SC) decoder. High errors are related to the so-called stopping sets of the underlying graphs. we discuss the pros and cons of BP decoder over SC decoder for various code lengths. / Thesis (Master, Electrical & Computer Engineering) -- Queen's University, 2012-01-31 17:10:59.955

[pt] Esta dissertação apresenta uma técnica de puncionamento orientada pela polarização para o projeto de códigos polares puncionados. A estratégia de puncionamento proposta consiste em reduzir a matriz geradora relacionando seu índice de linha com o princípio da polarização do canal. Os códigos puncionados construídos com base na polarização do canal são então considerados para a decodificação por cancelamento sucessivos (SC) com os bits perfurados conhecidos tanto no codificador como no decodificador. A Distância de Espectro (SD) e a Distância de Espectro Conjunta (JSD) são então utilizadas para análise de desempenho. Os resultados das simulações mostram que os códigos polares puncionados propostos superam os códigos polares puncionados existentes. / [en] This thesis presents a polarization-driven puncturing technique for the design of punctured polar codes. The proposed puncturing strategy consists of reducing the generator matrix by relating its row index based on the channel polarization principle. The punctured codes constructed based on channel polarization are then considered with successive cancellation (SC) decoding and punctured bits known to both the encoder and the decoder. The Spectrum Distance (SD) and the Joint Spectrum Distance (JSD) are then used to performance analysis. Simulation results show that the proposed punctured polar codes outperform existing punctured polar codes.

This dissertation has its focus on two different topics: 1. non-asymptotic analysis of polar codes as a new paradigm in error correcting codes with very promising features, and 2. network properties for wireless networks of practical size. In its first part, we investigate properties of polar codes that can be potentially useful in real-world applications. We start with analyzing the performance of finite-length polar codes over the binary erasure channel (BEC), while assuming belief propagation (BP) as the decoding method. We provide a stopping set analysis for the factor graph of polar codes, where we find the size of the minimum stopping set. Our analysis along with bit error rate (BER) simulations demonstrates that finite-length polar codes show superior error floor performance compared to the conventional capacity-approaching coding techniques. Motivated by good error floor performance, we introduce a modified version of BP decoding while employing a guessing algorithm to improve the BER performance. Each application may impose its own requirements on the code design. To be able to take full advantage of polar codes in practice, a fundamental question is which practical requirements are best served by polar codes. For example, we will see that polar codes are inherently well-suited for rate-compatible applications and they can provably achieve the capacity of time-varying channels with a simple rate-compatible design. This is in contrast to LDPC codes for which no provably universally capacity-achieving design is known except for the case of the erasure channel. This dissertation investigates different approaches to applications such as UEP, rate-compatible coding, and code design over parallel sub-channels (non-uniform error correction). Furthermore, we consider the idea of combining polar codes with other coding schemes, in order to take advantage of polar codes' best properties while avoiding their shortcomings. Particularly, we propose, and then analyze, a polar code-based concatenated scheme to be used in Optical Transport Networks (OTNs) as a potential real-world application The second part of the dissertation is devoted to the analysis of finite wireless networks as a fundamental problem in the area of wireless networking. We refer to networks as being finite when the number of nodes is less than a few hundred. Today, due to the vast amount of literature on large-scale wireless networks, we have a fair understanding of the asymptotic behavior of such networks. However, in real world we have to face finite networks for which the asymptotic results cease to be valid. Here we study a model of wireless networks, represented by random geometric graphs. In order to address a wide class of the network's properties, we study the threshold phenomena. Being extensively studied in the asymptotic case, the threshold phenomena occurs when a graph theoretic property (such as connectivity) of the network experiences rapid changes over a specific interval of the underlying parameter. Here, we find an upper bound for the threshold width of finite line networks represented by random geometric graphs. These bounds hold for all monotone properties of such networks. We then turn our attention to an important non-monotone characteristic of line networks which is the Medium Access (MAC) layer capacity, defined as the maximum number of possible concurrent transmissions. Towards this goal, we provide a linear time algorithm which finds a maximal set of concurrent non-interfering transmissions and further derive lower and upper bounds for the cardinality of the set. Using simulations, we show that these bounds serve as reasonable estimates for the actual value of the MAC-layer capacity.

<p>Two coding schemes based on polar codes are proposed for the multiple description (MD) problem. The first scheme is an adaptation of the one developed by \c{S}a\c{s}o\u{g}lu {\em et al.} for the multiple access channel to the MD setting. Specifically, it is shown that the scheme is able to achieve certain rate pairs on the dominant line of the achievable rate region determined by El Gamal and Cover (EGC) when the associated auxiliary random variables have different prime alphabet sizes. Different from polar coding for the multiple access channel considered by \c{S}a\c{s}o\u{g}lu {\em et al.}, the auxiliary random variables in the MD problem can be dependent, and their alphabet sizes play an unexpected important role in the construction. The second scheme is based on the idea of rate splitting. We show that it can achieve the entire EGC rate region. The effectiveness of the proposed polar coding schemes is verified by the experimental results.</p> / Master of Applied Science (MASc)

Polar codes, introduced by Arikan in 2009, have attracted considerable interest as an asymptotically capacity-achieving code with sufficient performance advantages to merit inclusion in the 5G standard. Polar codes are constructed directly from an explicit model of the communication channel, so their performance is dependent on a detailed understanding of the transmission environment. We partially remove a basic assumption in coding theory that channels are identical and independent by extending polar codes to several types of channels with memory, including periodic Markov processes and Information Regular processes. In addition, we consider modifications to the polar code construction so that the inclusion of a shared secret in the frozen set naturally produces encryption via one-time pad. We describe one such modification in terms of the achievable frozen sets which are compatible with the polar code automorphism group. We then provide a partial characterization of these frozen sets using an explicit construction for the Linear Extension Diameter of channel entropies. / Doctor of Philosophy / Efficient, reliable communication has become an essential component of modern society. Error-correcting codes allow for the use of redundant symbols to fix errors in transmission. While it has long been known that communication channels have an inherent capacity describing the optimal redundancy required for reliable transmission, explicit constructions which achieve this capacity have proved elusive. Our focus is the recently discovered family of polar codes, which are known to be asymptotically capacity-achieving. Polar codes also perform well enough in practice to merit inclusion in the 5G wireless standard shortly after their creation. The polarization process uses an explicit model of the channel and a recursive construction to concentrate errors in a few symbols (called the frozen set), which are then simply ignored. This reliance on an explicit channel model is problematic due to a long-standing assumption in coding theory that the probability of error in each symbol is identical and independent. We extend existing results to explore persistent sources of interference modelling environments such as nearby power lines or prolonged outages. While polar codes behave quite well in these new settings, some forms of memory can only be overcome using very long codewords. We next explore an application relating to secure communication, where messages must be recovered by a legitimate receiver but not by an eavesdropper. Polar codes behave quite well in this environment as well, as we can separately compute which symbols can be recovered by each party and use only those with the desired properties. We extend a recent result which proposes the use of a shared secret in the code construction to further complicate recovery by an eavesdropper. We consider several modifications to the construction of polar codes which allow the shared secret to be used for encryption in addition to the existing information theoretic use. We discover that this task is closely related to the unsolved problem of determining which symbols are in the frozen set for a particular channel. We conclude with partial results to this problem, including two choices of frozen set which are, in some sense, maximally separated.