Sub-4.7 Scaling Exponent of Polar Codes

Abstract

Polar codes approach channel capacity provably and empirically and are thereby a constituent code of the 5G standard. Compared to low-density parity-check codes, however, the performance of short-length polar codes have 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. The idea behind this improvement is that, instead of describing the relation between a channel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> and its children <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> ◻ and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> ⊛, we describe the relation between <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> and its grandchildren <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> ◻◻, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> ◻⊛, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> ⊛◻, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> ⊛⊛. By doing so, the evolution of channels becomes “less Markovian” and hence more tighter inequalities can be obtained.