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

TL;DR

This paper introduces sphere constraint based enumeration methods to exactly analyze the minimum weight distribution of polar and concatenated polar codes, enabling efficient enumeration for longer code lengths and code optimization.

Contribution

It proposes three novel enumeration methods leveraging sphere constraints and code structure to precisely compute the minimum weight distribution of polar codes and their concatenations.

Findings

SCREM can enumerate MWD of polar codes up to length 2^14

Proposed methods exactly enumerate all codewords in the MWD

PC-SCEM can optimize CRC-polar concatenated codes

Abstract

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 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 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.