Ergodic Theory Meets Polarization. II: A Foundation of Polarization Theory

Abstract

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 Arıkan 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 $\frac{1}{2}$, which is the exponent of quasigroup 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 $\frac{1}{2}$, which is the exponent of sequences of quasigroup operations.

33 pages. Accepted to IEEE Trans. Inform. Theory and presented in part at ISIT'15

References in corpus (1)

• Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations

Cited by in corpus (8)

• Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations
• Polar Codes' Simplicity, Random Codes' Durability
• Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs
• Continuity of Channel Parameters and Operations under Various DMC Topologies
• Fourier Analysis of MAC Polarization
• The Fractality of Polar and Reed-Muller Codes
• Modular Arithmetic Erasure Channels and Their Multilevel Channel Polarization
• Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions