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In successive cancellation (SC) polar decoding, an incorrect estimate of anyprior unfrozen bit may bring about severe error propagation in the followingdecoding, thus it is desirable to find out and correct an error as early aspossible. In this paper, we first construct a critical set $S$ of unfrozenbits, which with high probability (typically $>99\%$) includes the bit wherethe first error happens. Then we develop a progressive multi-level bit-flippingdecoding algorithm to correct multiple errors over the multiple-layer criticalsets each of which is constructed using the remaining undecoded subtreeassociated with the previous layer. The \emph{level} in fact indicates thenumber of \emph{independent} errors that could be corrected. We show that asthe level increases, the block error rate (BLER) performance of the proposedprogressive bit flipping decoder competes with the corresponding cyclicredundancy check (CRC) aided successive cancellation list (CA-SCL) decoder,e.g., a level 4 progressive bit-flipping decoder is comparable to the CA-SCLdecoder with a list size of $L=32$. Furthermore, the average complexity of theproposed algorithm is much lower than that of a SCL decoder (and is similar tothat of SC decoding) at medium to high signal to noise ratio (SNR).

Progressive Bit-Flipping Decoding of Polar Codes Over Layered Critical Sets