Re-proving Channel Polarization Theorems: An Extremality and Robustness Analysis

Abstract The general subject considered in this thesis is a recently discovered coding technique,polar coding, which is used to construct a class of error correction codes withunique properties. In his ground-breaking work, Arikan proved that this class ofcodes, called polar codes, achieve the symmetric capacity — the mutual informationevaluated at the uniform input distribution — of any stationary binary discretememoryless channel with low complexity encoders and decoders requiring in theorder of O(NlogN) operations in the block-length N. This discovery settledthe long standing open problem left by Shannon of finding low complexity codesachieving the channel capacity.Polar codes are not only appealing for being the first to ‘close the deal’. Incontrast to most of the existing coding schemes, polar codes admit an explicit lowcomplexity construction. In addition, for symmetric channels, the polar code con-struction is deterministic; the theoretically beautiful but practically limited “averageperformance of an ensemble of codes is good, so there must exist one particular codein the ensemble at least as good as the average” formalism of information theory isbypassed. Simulations are thus not necessary in principle for evaluating the errorprobability which is shown in a study by Telatar and Arikan to scale exponentially inthe square root of the block-length. As such, at the time of this writing, polar codesare appealing for being the only class of codes proved, and proved with mathematicalelegance, to possess all of these properties.Polar coding settled an open problem in information theory, yet opened plentyof challenging problems that need to be addressed. This novel coding scheme is apromising method from which, in addition to data transmission, problems such asdata compression or compressed sensing, which includes all types of measurementprocesses like the MRI or ultrasound, could benefit in terms of efficiency. To makethis technique fulfill its promise, the original theory has been, and should still be,extended in multiple directions. A significant part of this thesis is dedicated toadvancing the knowledge about this technique in two directions. The first oneprovides a better understanding of polar coding by generalizing some of the existingresults and discussing their implications, and the second one studies the robustnessof the theory over communication models introducing various forms of uncertaintyor variations into the probabilistic model of the channel.The idea behind the design of a polar code is a phenomenon called channelxi