On the scaling of Polar Codes: II. The behavior of un-polarized channels

TL;DR

This paper analyzes the behavior of unpolarized channels in polar codes over the binary erasure channel by establishing bounds on the rate at which sub-channels polarize, focusing on the probability of remaining unpolarized.

Contribution

It provides new upper and lower bounds on the escape rate of the Bhattacharyya process, quantifying the polarization speed of sub-channels.

Findings

Bounds on the exponent of unpolarized sub-channels

Quantitative characterization of the polarization process

Insights into the asymptotic behavior of the Bhattacharyya process

Abstract

We provide upper and lower bounds on the escape rate of the Bhattacharyya process corresponding to polar codes and transmission over the the binary erasure channel. More precisely, we bound the exponent of the number of sub-channels whose Bhattacharyya constant falls in a fixed interval $[a,b]$. Mathematically this can be stated as bounding the limit $\lim_{n \to \infty} \frac{1}{n} \ln \mathbb{P}(Z_n \in [a,b])$, where $Z_n$ is the Bhattacharyya process. The quantity $\mathbb{P}(Z_n \in [a,b])$ represents the fraction of sub-channels that are still un-polarized at time $n$.