Finance

Combinatorics

Discrete mathematics

Physics

Pure mathematics

Block (permutation group theory)

Mathematics

Algorithm

Astronomy

Automorphism

Code word

Order (exchange)

Block code

Economics

Set (abstract data type)

Programming language

List decoding

Code (set theory)

Polar code

Group (periodic table)

Polar

Automorphism group

Computer science

Quantum mechanics

Decoding methods

Affine group

Concatenated error correction code

Affine transformation

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.

On the Automorphism Group of Polar Codes