Enumeration of Minimum Weight Codewords of Pre-Transformed Polar Codes by Tree Intersection

Author:

Andreas Zunker, Marvin Geiselhart, Stephan ten Brink

Keyword:

Computer Science, Information Theory, Information Theory (cs.IT)

journal:

date:

2023-11-29 00:00:00

Abstract

Pre-transformed polar codes (PTPCs) form a class of codes that perform close to the finite-length capacity bounds. The minimum distance and the number of minimum weight codewords are two decisive properties for their performance. In this work, we propose an efficient algorithm to determine the number of minimum weight codewords of general PTPCs, which eliminates all redundant visits of nodes of the search tree, reducing the computational complexity from state-of-the-art algorithms typically by several orders of magnitude. The algorithm is demonstrated for randomly pre-transformed Reed-Muller (RM) codes and polarization-adjusted convolutional (PAC) codes. Optimal convolutional polynomials for PAC codes are designed, minimizing the number of minimum weight codewords.

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