Polar Codes Do Not Have Many Affine Automorphisms

TL;DR

This paper investigates the symmetry properties of Arikan's polar codes, revealing that their affine automorphism group is limited and does not significantly extend beyond lower-triangular permutations, impacting permutation decoding strategies.

Contribution

The study demonstrates that the affine automorphism group of Arikan's polar codes is asymptotically small, constraining the potential for permutation decoding improvements.

Findings

Affine automorphisms are mostly limited to lower-triangular permutations.

The automorphism group size does not grow significantly with code length.

Implications for permutation decoding are constrained by these structural properties.

Abstract

Polar coding solutions demonstrate excellent performance under the list decoding that is challenging to implement in hardware due to the path sorting operations. As a potential solution to this problem, permutation decoding recently became a hot research topic. However, it imposes more constraints on the code structure. In this paper, we study the structural properties of Arikan's polar codes. It is known that they are invariant under lower-triangular affine permutations among others. However, those permutations are not useful in the context of permutation decoding. We show that, unfortunately, the group of affine automorphisms of Arikan's polar codes asymptotically cannot be much bigger than the group of lower-triangular permutations.