Sub-4.7 Scaling Exponent of Polar Codes

TL;DR

This paper improves the upper bound on the scaling exponent of polar codes, which measures the speed of channel polarization, from 4.714 to 4.63, aiding the design of more efficient short-length polar codes.

Contribution

It provides a tighter upper estimate of the scaling exponent of polar codes, enhancing understanding of their polarization speed at short lengths.

Findings

Lowered the overestimate of the scaling exponent to 4.63

Improved understanding of polarization speed for short-length polar codes

Supports better design of polar codes for practical applications

Abstract

Polar code visibly approaches channel capacity in practice and is thereby a constituent code of the 5G standard. Compared to low-density parity-check code, however, the performance of short-length polar code has rooms for improvement that could hinder its adoption by a wider class of applications. As part of the program that addresses the performance issue at short length, it is crucial to understand how fast binary memoryless symmetric channels polarize. A number, called scaling exponent, was defined to measure the speed of polarization and several estimates of the scaling exponent were given in literature. As of 2022, the tightest overestimate is 4.714 made by Mondelli, Hassani, and Urbanke in 2015. We lower the overestimate to 4.63.