On the Weight Spectrum of Pre-Transformed Polar Codes

Polar codes are the first class of channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels (B-DMC) with efficient encoding and decoding algorithms. But the weight spectrum of polar codes is relatively poor compared to Reed-Muller (RM) codes, which degrades their maximum-likehood (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.