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

TL;DR

This paper establishes a fundamental foundation for polarization theory using ergodic theory, providing necessary and sufficient conditions for binary operations to always induce polarization, and analyzing their polarization rates and applications to multiple access channels.

Contribution

It offers a complete characterization of polarizing binary operations based on ergodic theory, and extends the analysis to polarization of multiple access channels with rate bounds.

Findings

A binary operation is polarizing iff it is uniformity preserving and its right-inverse is strongly ergodic.

The polarization exponent of any polarizing operation cannot exceed 1/2.

A sequence of binary operations is MAC-polarizing iff each operation is polarizing.

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 Ar{\i}kan 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.