Group Properties of Polar Codes for Automorphism Ensemble Decoding

TL;DR

This paper analyzes the automorphism group of polar codes to enhance automorphism ensemble decoding, providing algebraic insights, classification of automorphisms, and methods to identify permutations that improve decoding performance.

Contribution

It offers a novel algebraic analysis of polar code automorphisms, classifies automorphisms for permutation decoding, and proposes a method to list all automorphisms affecting AE decoding outcomes.

Findings

Automorphism group structure of polar codes characterized.

Method to identify automorphisms leading to different decoding candidates.

Numerical results showing improved error correction with AE decoding.

Abstract

In this paper, we propose an analysis of the automorphism group of polar codes, with the scope of designing codes tailored for automorphism ensemble (AE) decoding. We prove the equivalence between the notion of decreasing monomial codes and the universal partial order (UPO) framework for the description of polar codes. Then, we analyze the algebraic properties of the affine automorphisms group of polar codes, providing a novel description of its structure and proposing a classification of automorphisms providing the same results under permutation decoding. Finally, we propose a method to list all the automorphisms that may lead to different candidates under AE decoding; by introducing the concept of redundant automorphisms, we find the maximum number of permutations providing possibly different codeword candidates under AE-SC, proposing a method to list all of them. A numerical analysis of the error correction performance of AE algorithm for the decoding of polar codes concludes the paper.