  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.
