In this paper, we prove that any pre-transformation with an upper-triangular matrix (including cyclic redundancy check (CRC), parity-check (PC) and convolution code (CC) matrix) does not reduce the code minimum distance and an properly designed pre-transformation can reduce the number of codewords with the minimum distance. This explains that the pre-transformed polar codes can perform better than the Polar/RM codes.
@article{arxiv.1912.06359,
title = {On Pre-transformed Polar Codes},
author = {Bin Li and Huazi Zhang and Jiaqi Gu},
journal= {arXiv preprint arXiv:1912.06359},
year = {2019}
}