On automorphism groups of polar codes

TL;DR

This paper characterizes the automorphism groups of Polar codes, linking their structure to Reed-Muller codes, and provides a reduction theorem to simplify automorphism group determination.

Contribution

It introduces a reduction theorem for automorphism groups of Polar codes and classifies their automorphism groups for Reed-Muller-based constructions.

Findings

Reduction theorem for automorphism groups of Polar codes

Complete classification for Reed-Muller constructed Polar codes

Enhanced understanding of code automorphisms for decoding improvements

Abstract

Over the past years, Polar codes have arisen as a highly effective class of linear codes, equipped with a decoding algorithm of low computational complexity. This family of codes share a common algebraic formalism with the well-known Reed-Muller codes, which involves monomial evaluations. As useful algebraic codes, more specifically known as decreasing monomial codes, a lot of decoding work has been done on Reed-Muller codes based on their rich code automorphisms. In 2021, a new permutation group decoder, referred to as the automorphism ensemble (AE) decoder, was introduced. This decoder can be applied to Polar codes and has been shown to produce similar decoding effects. However, identifying the right set of code automorphisms that enhance decoding performance for Polar codes remains a challenging task. This paper aims to characterize the full automorphism group of Polar codes. We will prove a reduction theorem that effectively reduces the problem of determining the full automorphism group of arbitrary random Polar codes to that of a specified class of Polar codes. Besides, we give exact classification of the full automorphism groups of families of Polar codes that are constructed using the Reed-Muller codes.