Capacity-Achievability of Polar Codes and Precoded Polar Codes under List Decoding: They are Good both Practically and Theoretically

TL;DR

This paper proves that polar codes and precoded polar codes with list decoding can theoretically achieve the symmetric capacity of binary-input discrete memoryless channels, confirming their practical effectiveness.

Contribution

It formalizes the success of list decoding for polar and precoded polar codes, establishing their capacity-achieving properties in theory.

Findings

Polar codes under list decoding achieve symmetric capacity.

Precoded polar codes also achieve capacity under certain conditions.

Theoretical validation of practical decoding performance.

Abstract

Polar codes under successive cancellation decoding proposed by Ar{\i}kan provably achieve the symmetric capacity of any given binary-input discrete memoryless channel. The successive cancellation list decoder for polar codes was described by Tal and Vardy as a generalization of the successive cancellation decoder of Ar{\i}kan. The performance of the successive cancellation list decoder is encouraging in practice. In this paper, we formalize the successive cancellation list decoder in our notation and prove that polar codes under successive cancellation list decoding achieve the symmetric capacity of any given binary-input discrete memoryless channel in theory as well. We also formalize the polar codes with CRC precoding of Tal and Vardy. In fact, we propose a family of more general codes, namely, precoded polar codes and prove that precoded polar codes under successive cancellation list decoding can achieve the symmetric capacity of any given binary-input discrete memoryless channel under some conditions.