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In this paper, the minimum weight distributions (MWDs) of polar codes andconcatenated polar codes are exactly enumerated according to the distanceproperty of codewords. We first propose a sphere constraint based enumerationmethod (SCEM) to analyze the MWD of polar codes with moderate complexity. TheSCEM exploits the distance property that all the codewords with the identicalHamming weight are distributed on a spherical shell. Then, based on the SCEMand the Plotkin's construction of polar codes, a sphere constraint basedrecursive enumeration method (SCREM) is proposed to recursively calculate theMWD 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-checkequations of outer codes. Moreover, due to the distance property of codewords,the proposed three methods can exactly enumerate all the codewords belonging tothe MWD. The enumeration results show that the SCREM can enumerate the MWD ofpolar codes with code length up to $2^{14}$ and the PC-SCEM can be used tooptimize CRC-polar concatenated codes.

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