On the Automorphism Group of Polar Codes

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 s...

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Bibliographic Details
Published in2021 IEEE International Symposium on Information Theory (ISIT) pp. 1230 - 1235
Main Authors Geiselhart, Marvin, Elkelesh, Ahmed, Ebada, Moustafa, Cammerer, Sebastian, ten Brink, Stephan
Format Conference Proceeding
LanguageEnglish
Published IEEE 12.07.2021
Subjects
Complexity theory
Decoding
Performance gain
Polar codes
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Summary: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.
DOI:10.1109/ISIT45174.2021.9518184