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Person# Mine Alsan

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Polar code (coding theory)

In information theory, a polar code is a linear block error-correcting code. The code construction is based on a multiple recursive concatenation of a short kernel code which transforms the physical

Binary symmetric channel

A binary symmetric channel (or BSCp) is a common communications channel model used in coding theory and information theory. In this model, a transmitter wishes to send a bit (a zero or a one), and th

Communication channel

A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and compute

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We study the extremality of the binary erasure channel and the binary symmetric channel for Gallager's reliability function E-0 of binary input discrete memoryless channels evaluated under the uniform input distribution from the aspect of channel polarization. In particular, we show that amongst all binary discrete memoryless channels of a given E-0(rho) value, for a fixed rho >= 0, the binary erasure channel and the binary symmetric channel are extremal in the evolution of E-0 under the one-step polarization transformations.

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We prove that channel combining and splitting via Arikan's polarization transformation improves Gallager's reliability function E-0 for binary input channels. In this sense, polarization creates E-0. This observation gives yet another justification as to why the polar transform yields capacity achieving and low complexity codes: the improvement in E-0 translates to an improvement in complexity-error-probability trade-off. In analyzing polar codes, one examines auxiliary random processes that follow the evolution of information measures as an underlying communication channel undergoes a sequence of transformations. The conclusion of this paper shows that the E-0 process associated to such an analysis is a submartingale.

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 with unique properties. In his ground-breaking work, Arikan proved that this class of codes, called polar codes, achieve the symmetric capacity --- the mutual information evaluated at the uniform input distribution ---of any stationary binary discrete memoryless channel with low complexity encoders and decoders requiring in the order of $O(N\log N)$ operations in the block-length $N$. This discovery settled the long standing open problem left by Shannon of finding low complexity codes achieving the channel capacity. Polar codes are not only appealing for being the first to 'close the deal'. In contrast to most of the existing coding schemes, polar codes admit an explicit low complexity construction. In addition, for symmetric channels, the polar code construction is deterministic; the theoretically beautiful but practically limited "average performance of an ensemble of codes is good, so there must exist one particular code in the ensemble at least as good as the average'' formalism of information theory is bypassed. Simulations are thus not necessary in principle for evaluating the error probability which is shown in a study by Telatar and Arikan to scale exponentially in the square root of the block-length. As such, at the time of this writing, polar codes are appealing for being the only class of codes proved, and proved with mathematical elegance, to possess all of these properties. Polar coding settled an open problem in information theory, yet opened plenty of challenging problems that need to be addressed. This novel coding scheme is a promising method from which, in addition to data transmission, problems such as data compression or compressed sensing, which includes all types of measurement processes like the MRI or ultrasound, could benefit in terms of efficiency. To make this 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 to advancing the knowledge about this technique in two directions. The first one provides a better understanding of polar coding by generalizing some of the existing results and discussing their implications, and the second one studies the robustness of the theory over communication models introducing various forms of uncertainty or variations into the probabilistic model of the channel. See the fulltext of the thesis for the complete abstract.