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Publication# Ergodic Theory Meets Polarization I: A Foundation of Polarization Theory

Résumé

An open problem in polarization theory is to determine the binary operations that always lead to polarization when they are used in Arıkan style constructions. This paper solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. The characterization is given in terms of a new mathematical framework that we introduce. We show that a binary operation is polarizing if and only if its inverse is strongly ergodic.

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Problème ouvert

En mathématiques, un problème ouvert est une question qui n'a pas été résolue ou une conjecture qui n'a pas été prouvée.
En didactique des mathématiques, un problème ouvert est une question posée

Ergodicity

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in

Opération binaire

Les opérations en codage binaire sont traitées à l'article Fonction logique.
En mathématiques, une opération binaire est une opération à deux arguments ou opérandes.
C'est le cas notamment des lois

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An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arikan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (first) part of this paper introduces the mathematical framework that we will use in the second part to characterize the polarizing operations. We define uniformity preserving, irreducible, ergodic, and strongly ergodic operations, and we study their properties. The concepts of a stable partition and the residue of a stable partition are introduced. We show that an ergodic operation is strongly ergodic if and only if all its stable partitions are their own residues. We also study the products of binary operations and the structure of their stable partitions. We show that the product of a sequence of binary operations is strongly ergodic if and only if all the operations in the sequence are strongly ergodic. In the second part of this paper, we provide a foundation of polarization theory based on the ergodic theory of binary operations that we develop in this part.

An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arikan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (second) part provides a foundation of polarization theory based on the ergodic theory of binary operations which we developed in the first part. We show that a binary operation is polarizing if and only if it is uniformity preserving and its right-inverse is strongly ergodic. The rate of polarization of single user channels is studied. It is shown that the exponent of any polarizing operation cannot exceed 1/2, which is the exponent of quasi-group operations. We also study the polarization of multiple access channels (MAC). In particular, we show that a sequence of binary operations is MAC-polarizing if and only if each binary operation in the sequence is polarizing. It is shown that the exponent of any MAC-polarizing sequence cannot exceed 1/2, which is the exponent of sequences of quasi-group operations.

Information theory is the field in which we study the fundamental limitations of communication. Shannon proved in 1948 that there exists a maximum rate, called capacity, at which we can reliably communicate information through a given channel. However, Shannon did not provide an explicit construction of a practical coding scheme that achieves the capacity. Polar coding, invented by Arikan, is the first low-complexity coding technique that achieves the capacity of binary-input memoryless symmetric channels. The construction of these codes is based on a phenomenon called polarization. The study of polar codes and their generalization to arbitrary channels is the subject of polarization theory, a subfield of information and coding theories. This thesis consists of two parts. In the first part, we provide solutions to several open problems in polarization theory. The first open problem that we consider is to determine the binary operations that always lead to polarization when they are used in Arikan-style constructions. In order to solve this problem, we develop an ergodic theory for binary operations. This theory is used to provide a necessary and sufficient condition that characterizes the polarizing binary operations, both in the single-user and the multiple-access settings. We prove that the exponent of a polarizing binary operation cannot exceed 1/2. Furthermore, we show that the exponent of an arbitrary quasigroup operation is exactly 1/2. This implies that quasigroup operations are among the best polarizing binary operations. One drawback of polarization in the multiple-access setting is that it sometimes induces a loss in the symmetric capacity region of a given multiple-access channel (MAC). An open problem in MAC polarization theory is to determine all the MACs that do not lose any part of their capacity region by polarization. Using Fourier analysis, we solve this problem by providing a single-letter necessary and sufficient condition that characterizes all these MACs in the general setting where we have an arbitrary number of users, and each user uses an arbitrary Abelian group operation on his input alphabet. We also study the polarization of classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation, and an Arikan-style transformation is applied using this operation. We show that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels that project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple-access channels (cq-MAC). In the second part of this thesis, we investigate several problems that are related to three orderings of communication channels: degradedness, input-degradedness, and the Shannon ordering. We provide several characterizations for the input-degradedness and the Shannon ordering. Two channels are said to be equivalent if they are degraded from each other. Input-equivalence and Shannon-equivalence between channels are similarly defined. We construct and study several topologies on the quotients of the spaces of discrete memoryless channels (DMC) by the equivalence, the input-equivalence and the Shannon-equivalence relations. Finally, we prove the continuity of several channel parameters and operations under various DMC topologies.