Fast Polarization for Processes with Memory

TL;DR

This paper demonstrates that the rate of polarization for processes with memory modeled by ergodic finite-state Markov chains matches that of memoryless processes, ensuring fast polarization in these more complex settings.

Contribution

It provides a complete characterization of the polarization rate for processes with memory, extending the understanding from memoryless models to Markovian processes.

Findings

Polarization rate for processes with memory matches the memoryless case.

Fast polarization applies to high and low entropy sets in Markovian processes.

The results hold for ergodic finite-state Markov chain models.

Abstract

Fast polarization is crucial for the performance guarantees of polar codes. In the memoryless setting, the rate of polarization is known to be exponential in the square root of the block length. A complete characterization of the rate of polarization for models with memory has been missing. Namely, previous works have not addressed fast polarization of the high entropy set under memory. We consider polar codes for processes with memory that are characterized by an underlying ergodic finite-state Markov chain. We show that the rate of polarization for these processes is the same as in the memoryless setting, both for the high and for the low entropy sets.