On automorphism groups of polar codes
Jicheng Ma, Guiying Yan
TL;DR
The paper addresses the problem of characterizing the full automorphism group of polar codes, including non-affine symmetries, by exploiting their link to decreasing monomial codes and the AE decoding paradigm. It introduces a reduction theorem that expresses the automorphism group of a polar code of length $2^n$ in terms of the automorphism group of a base code of length $2^m$, via ${\rm Aut}(\mathcal{P}_n) \cong ( {\rm S}_{2^{n-m}} )^{2^m} : G$ with $G \cong {\rm Aut}(\mathcal{P}_m)$, thereby reducing the problem to a smaller instance. For polar codes derived from Reed-Muller codes, the paper provides explicit classifications: when $2\le r\le n-3$, ${\rm Aut}(\mathcal{P}_r)$ is affine, taking the form $({\mathbb Z}_2)^{(n-m)(m+1)}: ({\rm AGL}(m,2) \times {\rm GL}(n-m,2))$; for $r=1$, it is $({\mathbb Z}_2)^{4(n-2)}: (S_4 \times {\rm GL}(n-2,2))$, and the case $r=n-2$ admits non-affine automorphisms. These results jointly map out when non-affine symmetries occur and connect to earlier affine-only findings, enabling more informed use of automorphisms in AE-based decoding. Overall, the work advances the structural understanding of polar-code symmetries and provides concrete group-theoretic tools for decoding approaches that leverage automorphisms.
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.