  Although iterative decoding of polar codes has recently made huge progress
based on the idea of permuted factor graphs, it still suffers from a
non-negligible performance degradation when compared to state-of-the-art
CRC-aided successive cancellation list (CA-SCL) decoding. In this work, we show
that iterative decoding of polar codes based on the belief propagation list
(BPL) algorithm can approach the error-rate performance of CA-SCL decoding and,
thus, can be efficiently used for decoding the standardized 5G polar codes.
Rather than only utilizing the cyclic redundancy check (CRC) as a stopping
condition (i.e., for error-detection), we also aim to benefit from the
error-correction capabilities of the outer CRC code. For this, we develop two
distinct soft-decision CRC decoding algorithms: a Bahl-Cocke-Jelinek-Raviv
(BCJR)-based approach and a sum product algorithm (SPA)-based approach.
Further, an optimized selection of permuted factor graphs is analyzed and shown
to reduce the decoding complexity significantly. Finally, we benchmark the
proposed CRC-aided belief propagation list (CA-BPL) to state-of-the-art 5G
polar codes under CA-SCL decoding and, thereby, showcase an error-rate
performance not just close to the CA-SCL but also close to the maximum
likelihood (ML) bound as estimated by ordered statistic decoding (OSD).
