Design and Analysis of Communication Schemes via Polar Coding
Gulcu, Talha Cihad
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Polar codes, introduced by Arikan in 2009, gave the first solution to the problem of designing explicit coding schemes that attain Shannon capacity of several basic models of communication channels. This discovery made it possible to attain theoretical limits of communication in a number of other problems of data compression and multi-user communication as well as provided new perspectives on extremal configurations of some discrete-time random walks. This thesis is devoted to the design of communication protocols for several basic information-theoretic problems as well to the problem of efficient construction of polar codes. In the first part we consider the problem of optimizing the amount of data transmit- ted between two terminals performing interactive computation of a function. Information- theoretic limits for one model of interactive computation were found in recent literature. We consider the distributed source coding problem that arises in the analysis of this model, designing a polar coding scheme that serves the basis for the distributed computation. As a result, it becomes possible to attain the smallest possible rate of data exchange between the terminals using an explicit protocol of encoding and data exchange that supports reli- able computation of the function by both parties. We also extend our considerations to a multi-terminal variation of this problem. Secondly, we turn to the problem of communication between two parties over a link observed by an adversary, known as the “wiretap channel.” Explicit capacity-achieving schemes for various models of the wiretap channel have received significant attention in recent literature. In this work, we address the general model of the channel, removing the constraints on the channels adopted in the earlier works. We show that secrecy capacity of the wiretap channel under a “strong secrecy constraint” can be achieved using an ex- plicit scheme based on polar codes. We also extend our construction to the case of the broadcast channel with confidential messages due to Csiszar and Korner, achieving the entire capacity region of this communication model. In the last part of the thesis we consider the problem of efficient construction of polar codes. While Arıkan’s scheme is explicit, his original proposal suffers from high construction complexity which grows exponentially with the number of evolution steps. An approximation procedure for binary-input channels was proposed and analyzed in the literature. Here we propose and study a construction algorithm for polar codes with arbitrarily-sized input alphabets. We establish a complexity estimate of the algorithm and derive an estimate of the approximation error that ensues from its use. The approximation error reduces the gap to the recently established lower bound for this type of algorithms. The validity of the proposed algorithm is supported by experimental results.