Progressive Bit-Flipping Decoding of Polar Codes: A Critical-Set Based Tree Search Approach

In successive cancellation (SC) polar decoding, an incorrect estimate of any prior unfrozen bit may bring about severe error propagation in the following decoding, and thus it is desirable to find out and correct an error as early as possible. In this paper, we investigate a progressive bit-flipping decoder which corrects at most <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>-<italic>independent</italic> errors in SC decoding. In particular, we first study the distribution of the first error position in SC decoding, and a critical set which with high probability includes the bit where the first error occurs regardless of the channel realizations is proposed. Second, a progressive bit-flipping decoding algorithm is proposed based on a search tree, which is established with a modified critical set in a progressive manner. The maximum <italic>level</italic> of the search tree is shown to coincide well with the number of independent errors that could be corrected. On this basis, the lower bound on BLER performance of a progressive bit-flipping decoder which corrects at most <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> errors is derived, and we show the bound can be tightly achieved by the proposed algorithm for some <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. Moreover, an early-terminated bit-flipping (ET-Bit-Flipping) decoder is proposed to reduce the computational complexity and decoding latency of the original progressive bit-flipping scheme. Finally, numerical results show that the proposed ET-bit-flipping decoders can provide almost the same BLER performance as the state-of-the-art cyclic redundancy check-aided SC list decoders, with an average computational complexity and decoding latency similar to that of the SC decoder at medium to a high SNR regime.