  The recently introduced polar codes constitute a breakthrough in coding
theory due to their capacityachieving property. This goes hand in hand with a
quasilinear construction, encoding, and successive cancellation list decoding
procedures based on the Plotkin construction. The decoding algorithm can be
applied with slight modifications to Reed-Muller or eBCH codes, that both
achieve the capacity of erasure channels, although the list size needed for
good performance grows too fast to make the decoding practical even for
moderate block lengths. The key ingredient for proving the capacity-achieving
property of Reed-Muller and eBCH codes is their group of symmetries. It can be
plugged into the concept of Plotkin decomposition to design various permutation
decoding algorithms. Although such techniques allow to outperform the
straightforward polar-like decoding, the complexity stays impractical. In this
paper, we show that although invariance under a large automorphism group is
valuable in a theoretical sense, it also ensures that the list size needed for
good performance grows exponentially. We further establish the bounds that
arise if we sacrifice some of the symmetries. Although the theoretical analysis
of the list decoding algorithm remains an open problem, our result provides an
insight into the factors that impact the decoding complexity.
