Sphere Constraint based Enumeration Methods to Analyze the Minimum Weight Distribution of Polar Codes

In this paper, the minimum weight distributions (MWDs) of polar codes and concatenated polar codes are exactly enumerated according to the distance property of codewords. We first propose a sphere constraint based enumeration method (SCEM) to analyze the MWD of polar codes with moderate complexity. The SCEM exploits the distance property that all the codewords with the identical Hamming weight are distributed on a spherical shell. Then, based on the SCEM and the Plotkins construction of polar codes, a sphere constraint based recursive enumeration method (SCREM) is proposed to recursively calculate the MWD with a lower complexity. Finally, we propose a parity-check SCEM (PC-SCEM) to analyze the MWD of concatenated polar codes by introducing the parity-check equations of outer codes. Moreover, due to the distance property of codewords, the proposed three methods can exactly enumerate all the codewords belonging to the MWD. The enumeration results show that the SCREM can enumerate the MWD of polar codes with code length up to $2^{14}$ and the PC-SCEM can be used to optimize CRC-polar concatenated codes.