A Method to Reduce the Complexity of Computing the Complete Weight Distribution of Polar Codes

TL;DR

This paper introduces an algebraic method leveraging group theory to significantly reduce the computational complexity of determining the complete weight distribution of polar codes, which is vital for their performance analysis.

Contribution

It defines a new subgroup within the lower-triangular affine group to identify more cosets with identical weight distributions, improving computational efficiency.

Findings

Complexity reduction exceeds several times in most cases.

The algebraic structure enables transitive group action on coset sets.

Enhanced method over previous approaches.

Abstract

The code spectrum of polar codes is crucial to the performance of polar codes. Based on the lower-triangular affine group (LTA) of decreasing monomial codes and the one-variable descendance (ovd) relation, we define a new subgroup of LTA which can find more cosets with the same weight distribution. Using this algebraic structure, we further reduce the complexity by proofing the group action on a coset set is transitive. Our method is an enhanced version of previous research, and the complexity of most cases can be reduced exceeding several times.