On the Distribution of Partially Symmetric Codes for Automorphism Ensemble Decoding

TL;DR

This paper investigates the distribution and properties of Partially-Symmetric Reed-Muller codes for automorphism ensemble decoding, demonstrating their existence, favorable permutation properties, and superior error correction for short codes.

Contribution

It proves the existence of PS-RM codes for most lengths up to 256, analyzes their permutation groups, and shows they outperform existing polar code constructions in error correction and latency.

Findings

PS-RM codes exist for almost all dimensions up to length 256.

Valuable permutations for AE decoding always exist in these codes.

PS-RM codes outperform state-of-the-art in error correction and decoding latency for short lengths.

Abstract

Automorphism Ensemble (AE) decoding has recently drawn attention as a possible alternative to list decoding of polar codes. In this letter, we investigate the distribution of Partially-Symmetric Reed-Muller (PS-RM) codes, a family of polar codes yielding good performances under AE decoding. We prove the existence of these codes for almost all code dimensions for code lengths $N\leq 256$. Moreover, we analyze the absorption group of this family of codes under SC decoding, proving that valuable permutations in AE decoding always exist. Finally, we experimentally show that PS-RM codes can outperform state-of-the-art polar-code-construction algorithms in terms of error-correction performance for short code lengths, while reducing decoding latency.