List Decoding of Polar Codes

• Published in: IEEE transactions on information theory Vol. 61; no. 5; pp. 2213 - 2226
• Main Authors: Tal, Ido, Vardy, Alexander
• Format: Journal Article
• Language: English
• Published: IEEE 01-05-2015
• Subjects: Arrays; Bit error rate; Complexity theory; List decoding; Maximum likelihood decoding; Parity check codes; polar codes; successive cancellation decoding; Vectors; successive cancellation decoding; List decoding; polar codes
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2015.2410251

We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arıkan. In the proposed list decoder, L decoding paths are considered concurrently at each decoding stage, where L is an integer parameter. At the end of the decoding process, the most likely among the L paths is selected as the single codeword at the decoder output. Simulations show that the resulting performance is very close to that of maximum-likelihood decoding, even for moderate values of L. Alternatively, if a genie is allowed to pick the transmitted codeword from the list, the results are comparable with the performance of current state-of-the-art LDPC codes. We show that such a genie can be easily implemented using simple CRC precoding. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit, and then uses a pruning procedure to discard all but the L most likely paths. However, straightforward implementation of this algorithm requires Ω(Ln 2 ) time, which is in stark contrast with the O(n log n) complexity of the original successive-cancellation decoder. In this paper, we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem: we devise an efficient, numerically stable, implementation of the proposed list decoder that takes only O(Ln logn) time and O(Ln) space.