The Fractality of Polar and Reed-Muller Codes

TL;DR

This paper explores the fractal properties of the index sets used in constructing polar and Reed-Muller codes, revealing their self-similarity and fractal dimensions in the infinite blocklength limit.

Contribution

It provides a detailed analysis of the fractal characteristics of the index sets for polar and Reed-Muller codes, a novel perspective in coding theory.

Findings

Index sets exhibit fractal properties such as self-similarity.

Hausdorff dimension of the index sets is characterized.

Lebesgue measure of the index sets is analyzed.

Abstract

The generator matrices of polar codes and Reed-Muller codes are obtained by selecting rows from the Kronecker product of a lower-triangular binary square matrix. For polar codes, the selection is based on the Bhattacharyya parameter of the row, which is closely related to the error probability of the corresponding input bit under sequential decoding. For Reed-Muller codes, the selection is based on the Hamming weight of the row. This work investigates the properties of the index sets pointing to those rows in the infinite blocklength limit. In particular, the Lebesgue measure, the Hausdorff dimension, and the self-similarity of these sets will be discussed. It is shown that these index sets have several properties that are common to fractals.