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