Universal Polar Codes for More Capable and Less Noisy Channels and Sources

We prove two results on the universality of polar codes for source coding and channel communication. First, we show that for any polar code built for a source $P_{X,Z}$ there exists a slightly modified polar code - having the same rate, the same encoding and decoding complexity and the same error rate - that is universal for every source $P_{X,Y}$ when using successive cancellation decoding, at least when the channel $P_{Y|X}$ is more capable than $P_{Z|X}$ and $P_X$ is such that it maximizes $I(X;Y) - I(X;Z)$ for the given channels $P_{Y|X}$ and $P_{Z|X}$. This result extends to channel coding for discrete memoryless channels. Second, we prove that polar codes using successive cancellation decoding are universal for less noisy discrete memoryless channels.Comment: 10 pages, 3 figure

Universal Polarization

A method to polarize channels universally is introduced. The method is based on combining two distinct channels in each polarization step, as opposed to Arikan's original method of combining identical channels. This creates an equal number of only two types of channels, one of which becomes progressively better as the other becomes worse. The locations of the good polarized channels are independent of the underlying channel, guaranteeing universality. Polarizing the good channels further with Arikan's method results in universal polar codes of rate 1/2. The method is generalized to construct codes of arbitrary rates. It is also shown that the less noisy ordering of channels is preserved under polarization, and thus a good polar code for a given channel will perform well over a less noisy one.Comment: Submitted to the IEEE Transactions on Information Theor

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

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, Ar{\i}kan 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 coding settled an open problem in information theory, yet opened plenty of challenging problems that need to be addressed. 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.Comment: Preview of my PhD Thesis, EPFL, Lausanne, 2014. For the full version, see http://people.epfl.ch/mine.alsan/publication

Universal Source Polarization and an Application to a Multi-User Problem

We propose a scheme that universally achieves the smallest possible compression rate for a class of sources with side information, and develop an application of this result for a joint source channel coding problem over a broadcast channel.Comment: to be presented at Allerton 201

On privacy amplification, lossy compression, and their duality to channel coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite-blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the $E_\gamma$ divergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with known relations between channel simulation and lossy source coding, this implies that privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Info. Theory 64, 577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a "curious duality" in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.Comment: v3: updated to include equivalence of the converse bound with smooth entropy formulations. v2: updated to include comparison with the one-shot bounds of arXiv:1706.03866. v1: 11 pages, 4 figure

Short Block-length Codes for Ultra-Reliable Low-Latency Communications

This paper reviews the state of the art channel coding techniques for ultra-reliable low latency communication (URLLC). The stringent requirements of URLLC services, such as ultra-high reliability and low latency, have made it the most challenging feature of the fifth generation (5G) mobile systems. The problem is even more challenging for the services beyond the 5G promise, such as tele-surgery and factory automation, which require latencies less than 1ms and failure rate as low as $10^{-9}$. The very low latency requirements of URLLC do not allow traditional approaches such as re-transmission to be used to increase the reliability. On the other hand, to guarantee the delay requirements, the block length needs to be small, so conventional channel codes, originally designed and optimised for moderate-to-long block-lengths, show notable deficiencies for short blocks. This paper provides an overview on channel coding techniques for short block lengths and compares them in terms of performance and complexity. Several important research directions are identified and discussed in more detail with several possible solutions.Comment: Accepted for publication in IEEE Communications Magazin

Principles of Physical Layer Security in Multiuser Wireless Networks: A Survey

This paper provides a comprehensive review of the domain of physical layer security in multiuser wireless networks. The essential premise of physical-layer security is to enable the exchange of confidential messages over a wireless medium in the presence of unauthorized eavesdroppers without relying on higher-layer encryption. This can be achieved primarily in two ways: without the need for a secret key by intelligently designing transmit coding strategies, or by exploiting the wireless communication medium to develop secret keys over public channels. The survey begins with an overview of the foundations dating back to the pioneering work of Shannon and Wyner on information-theoretic security. We then describe the evolution of secure transmission strategies from point-to-point channels to multiple-antenna systems, followed by generalizations to multiuser broadcast, multiple-access, interference, and relay networks. Secret-key generation and establishment protocols based on physical layer mechanisms are subsequently covered. Approaches for secrecy based on channel coding design are then examined, along with a description of inter-disciplinary approaches based on game theory and stochastic geometry. The associated problem of physical-layer message authentication is also introduced briefly. The survey concludes with observations on potential research directions in this area.Comment: 23 pages, 10 figures, 303 refs. arXiv admin note: text overlap with arXiv:1303.1609 by other authors. IEEE Communications Surveys and Tutorials, 201