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A Non-Asymptotic Approach to the Analysis of Communication Networks: From Error Correcting Codes to Network Properties

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.
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