On the efficiency of polar-like decoding for symmetric codes

TL;DR

This paper investigates the limitations of polar-like decoding algorithms for symmetric codes, showing that invariance under large automorphism groups leads to exponential growth in list size, impacting decoding practicality.

Contribution

It demonstrates that high symmetry in codes results in exponential list size growth for decoding, providing bounds and insights into the trade-offs between symmetry and decoding complexity.

Findings

Invariance under large automorphism groups causes exponential list size growth.

Reducing symmetries can mitigate list size growth, improving decoding practicality.

Theoretical bounds highlight the impact of symmetry on decoding complexity.

Abstract

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.