On the Finite Length Scaling of Ternary Polar Codes

TL;DR

This paper investigates the finite length scaling of ternary polar codes, demonstrating that the blocklength scales polynomially with a lower degree than previously known, which improves understanding of their efficiency.

Contribution

It provides a numerical method to compute a lower degree polynomial scaling law for ternary polar codes, extending the known results beyond the binary case.

Findings

Polynomial degree for blocklength scaling is significantly lower for ternary codes.

Numerical methods can effectively estimate the scaling law.

Conjectures extend these results to general prime alphabet sizes.

Abstract

The polarization process of polar codes over a ternary alphabet is studied. Recently it has been shown that the scaling of the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity. However, except for the binary case, the degree of the polynomial in the bound is extremely large. In this work, it is shown that a much lower degree polynomial can be computed numerically for the ternary case. Similar results are conjectured for the general case of prime alphabet size.