On Pre-transformed Polar Codes

TL;DR

This paper proves that certain pre-transformations do not reduce the minimum distance of polar codes and can improve their performance by reducing the number of minimum-distance codewords, explaining their enhanced effectiveness.

Contribution

It demonstrates that upper-triangular pre-transformations do not decrease minimum distance and can optimize code performance, providing theoretical insights into pre-transformed polar codes.

Findings

Pre-transformations with upper-triangular matrices do not reduce minimum distance.

Properly designed pre-transformations can decrease the number of minimum-distance codewords.

Pre-transformed polar codes outperform standard polar/RM codes in certain aspects.

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.