On Pre-transformed Polar Codes

Abstract

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.

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• A Soft Cancellation Decoder for Parity-Check Polar Codes
• On the Weight Spectrum of Pre-Transformed Polar Codes
• Complexity-Adaptive Maximum-Likelihood Decoding of Modified ${\mathbf{G}}_N$-Coset Codes