# The Penalty in Scaling Exponent for Polar Codes is Analytically Approximated by the Golden Ratio

## TL;DR

This paper analytically approximates the scaling exponent penalty in polar codes over BEC using percolation theory, revealing it is closely related to the golden ratio, providing a simple closed-form estimate.

## Contribution

It establishes a novel connection between the scaling exponent of polar codes and the golden ratio through percolation theory and analytical approximation.

## Key findings

- The scaling exponent is approximately 3.618, close to 2 plus the golden ratio.
- The analytical approximation matches numerical estimates within 0.028%.
- The penalty in scaling exponent can be effectively estimated by the golden ratio.

## Abstract

The polarization process of conventional polar codes in binary erasure channel (BEC) is recast to the Domany-Kinzel cellular automaton model of directed percolation in a tilted square lattice. Consequently, the former's scaling exponent, $\mu$, can be analogously expressed as the inverse of the percolation critical exponent, $\beta$. Relying on the vast percolation theory literature and the best known numerical estimate for $\beta$, the scaling exponent can be easily estimated as $\mu_{\text{num}}^{\text{perc}}\simeq1/0.276486(8)\simeq3.617$, which is only about $0.25\%$ away from the known exponent computation from coding theory literature based on numerical approximation, $\mu_{\text{num}}\simeq3.627$. Remarkably, this numerical result for the critical exponent, $\beta$, can be analytically approximated (within only $0.028\%$) leading to the closed-form expression for the scaling exponent $\mu\simeq2+\varphi=2+1.618\ldots\simeq3.618$, where $\varphi\triangleq(1+\sqrt{5})/2$ is the ubiquitous golden ratio. As the ultimate achievable scaling exponent is quadratic, this implies that the penalty for polar codes in BEC, in terms of the scaling exponent, can be very well estimated by the golden ratio, $\varphi$, itself.