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In this paper, we propose an analysis of the automorphism group of polarcodes, with the scope of designing codes tailored for automorphism ensemble(AE) decoding. We prove the equivalence between the notion of decreasingmonomial codes and the universal partial order (UPO) framework for thedescription of polar codes. Then, we analyze the algebraic properties of theaffine automorphisms group of polar codes, providing a novel description of itsstructure and proposing a classification of automorphisms providing the sameresults under permutation decoding. Finally, we propose a method to list allthe automorphisms that may lead to different candidates under AE decoding; byintroducing the concept of redundant automorphisms, we find the maximum numberof permutations providing possibly different codeword candidates under AE-SC,proposing a method to list all of them. A numerical analysis of the errorcorrection performance of AE algorithm for the decoding of polar codesconcludes the paper.

Group Properties of Polar Codes for Automorphism Ensemble Decoding