  Polar codes are the first class of channel codes achieving the symmetric
capacity of the binary-input discrete memoryless channels with efficient
encoding and decoding algorithms. But the weight spectrum of Polar codes is
relatively poor compared to RM codes, which degrades their ML performance.
Pre-transformation with an upper-triangular matrix (including cyclic redundancy
check (CRC), parity-check (PC) and polarization-adjusted convolutional (PAC)
codes), improves weight spectrum while retaining polarization. In this paper,
the weight spectrum of upper-triangular pre-transformed Polar Codes is
mathematically analyzed. In particular, we focus on calculating the number of
low-weight codewords due to their impact on error-correction performance.
Simulation results verify the accuracy of the analysis.
