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

As an important performance metric, the minimum weight distribution (MWD) determines the maximum-likelihood decoding performance. In this paper, we focus on the MWDs of polar codes and concatenated polar codes to analyze the performance and optimize the encoding structure. Existing enumeration methods cannot guarantee to get the MWD exactly and have high search complexity. In order to enumerate all the codewords belonging to the MWD, the sphere constraint property is applied, which means that the codewords with the identical Hamming weight are distribute on a spherical shell. Based on this, three enumeration methods are proposed in this paper. We first propose a sphere constraint based enumeration method (SCEM) to analyze the MWD of polar codes with moderate complexity. Then, according to the SCEM and the Plotkin's 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 sphere constraint property, the proposed three methods can exactly enumerate all the codewords with the minimum Hamming weight. The enumeration results show that the SCREM has about 10 ⁶ times complexity reduction compared with the existing methods at code length 128 and the PC-SCEM can be used to design high-performance CRC-polar concatenated codes.