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

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 ${\mu }_{\text{num}}^{\text{perc}}\simeq 1\mathrm{/}0.276486(8)\simeq 3.617$, which is only about $0.25\mathrm{\%}$ away from the known exponent computation from coding theory literature based on numerical approximation, ${\mu }_{\text{num}}\simeq 3.627$. Remarkably, this numerical result for the critical exponent, $\beta$, can be analytically approximated (within only $0.028\mathrm{\%}$) leading to the closed-form expression for the scaling exponent $\mu \simeq 2+\phi =2+1.618\dots \simeq 3.618$, where $\phi \triangleq (1+\sqrt{5})\mathrm{/}2$ 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, $\phi$, itself.