On the Automorphism Group of Polar Codes

TL;DR

This paper extends the known automorphism group of polar codes to a larger subgroup, the BLTA, and demonstrates its application in automorphism-based decoding with reduced complexity and competitive error performance.

Contribution

It introduces the block lower-triangular affine group as a larger automorphism subgroup for polar codes and provides an efficient algorithm to identify it.

Findings

BLTA is contained in the automorphism group of polar codes.

The proposed algorithm efficiently finds the automorphism group for given code sets.

Automorphism-based decoding achieves error rates comparable to SCL decoding with lower complexity.

Abstract

The automorphism group of a code is the set of permutations of the codeword symbols that map the whole code onto itself. For polar codes, only a part of the automorphism group was known, namely the lower-triangular affine group (LTA), which is solely based upon the partial order of the code's synthetic channels. Depending on the design, however, polar codes can have a richer set of automorphisms. In this paper, we extend the LTA to a larger subgroup of the general affine group (GA), namely the block lower-triangular affine group (BLTA) and show that it is contained in the automorphism group of polar codes. Furthermore, we provide a low complexity algorithm for finding this group for a given information/frozen set and determining its size. Most importantly, we apply these findings in automorphism-based decoding of polar codes and report a comparable error-rate performance to that of successive cancellation list (SCL) decoding with significantly lower complexity.