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

TL;DR

This paper introduces an efficient algorithm to count minimum weight codewords in pre-transformed polar codes, enabling better code design by directly considering minimum distance properties, demonstrated on RM and PAC codes.

Contribution

An efficient, reduced-complexity algorithm for enumerating minimum weight codewords in PTPCs, facilitating direct minimum distance optimization in code design.

Findings

Algorithm significantly reduces computational complexity.

Enables direct consideration of minimum distance in code design.

Optimized polynomials for PAC codes minimize minimum weight codewords.

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 for determining the number of minimum weight codewords of general PTPCs that eliminates all redundant visits to nodes of the search tree, thus reducing the computational complexity typically by several orders of magnitude compared to state-of-the-art algorithms. This reduction in complexity allows, for the first time, the minimum distance properties to be directly considered in the code design of PTPCs. The algorithm is demonstrated for randomly pre-transformed Reed-Muller (RM) codes and polarization-adjusted convolutional (PAC) codes. Furthermore, we design optimal polynomials for PAC codes with this algorithm, minimizing the number of minimum weight codewords.