Polar codes for channels with deletions

Abstract

A polar coding scheme for channels with deletions has been proposed recently, in which the information bits are pre-coded with CRC that helps to detect the location of deletions with high precision. Successive Cancellation (SC) decoding then treats these symbols as simple erasures. Given d as the number of deleted symbols, the decoding algorithm requires to check all (N d ) combinations of the deleted locations to find one that agrees with the CRC. This escalates the overall decoding complexity to O(Nd+1 log N), which is not practical even when d is a small number. In this paper, we propose an alternative decoding method for polar codes in presence of deletion errors. This method can be regarded as an extension of SC decoding to the deletion channel. The proposed algorithm is based on the recursive structure of polar codes and it directly adopts the outputs of deletion channel to perform decoding without any preprocessing. In other words, it is no longer required to check all (N d ) possible locations of the deletions. Instead, each node in the proposed polar decoder propagates its uncertainty about deletion pattern to the nodes in the next decoding layer. Eventually, with high probability, the correct deletion pattern becomes visible when the last polar bit-channel is decoded. The resulting decoding complexity is only O(d2 N log N), which scales polynomially rather than exponentially with the number of deletions.