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Pre-transformation with an upper-triangular matrix (including cyclicredundancy check (CRC), parity-check (PC) and polarization-adjustedconvolutional (PAC) codes) improves the weight spectrum of Reed-Muller (RM)codes and polar codes significantly. However, a theoretical analysis toquantify the improvement is missing. In this paper, we provide asymptoticanalysis on the number of low-weight codewords of the original andpre-transformed RM codes respectively, and prove that pre-transformationsignificantly reduces low-weight codewords, even in the order sense. For polarcodes, we prove that the average number of minimum-weight codewords does notincrease after pre-transformation. Both results confirm the advantages ofpre-transformation.

On the Weight Spectrum Improvement of Pre-transformed Reed-Muller Codes and Polar Codes