On the efficiency of polar-like decoding for symmetric codes
Communication Theory Laboratory
The recently introduced polar codes constitute a breakthrough in coding theory due to their capacity-achieving 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.
Kirill Ivanov (Student Member, IEEE) received the B.Sc. and M.Sc. degrees from St. Petersburg Polytechnic University, Russia, in 2015 and 2017, respectively. Currently he is working toward the Ph.D. degree at École polytechnique fédérale de Lausanne, Switzerland under the supervision of Prof. Rüdiger Urbanke. His research interests include wireless communications systems and coding theory, with focus on polar and Reed-Muller codes.