Affine Automorphism Group of Polar Codes

Abstract

The automorphism ensemble (AE) decoding framework for polar codes attracts much attention recently. It decodes multiple permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to successive cancellation list (SCL) decoding. However, the AE decoding framework is ineffective for permutations falling into the lower-triangular affine (LTA) automorphism group, as they are invariant under SC decoding. Therefore, the block lower-triangular affine (BLTA) group was discovered to achieve better AE decoding performance. However, the equivalence of the BLTA group and the complete affine automorphism group was unresolved. Additionally, some automorphisms in BLTA group are also SC-invariant, thus are redundant in AE decoding. In this paper, we prove that BLTA group coincides with the complete automorphisms of decreasing polar codes that can be formulated as affine transformations. Also, we find a necessary and sufficient condition related to the block lower-triangular structure of transformation matrices to identify SC-invariant automorphisms. Furthermore, We present an algorithm that efficiently identifies all SC-invariant affine automorphisms under specific constructions.