Approximate Weight Distribution of Polarization-Adjusted Convolutional (PAC) Codes

TL;DR

This paper introduces an approximate method to compute the weight distribution of PAC codes, enabling better code design and demonstrating their near-optimal performance for short block lengths.

Contribution

It develops a probabilistic approach to estimate weight distributions of PAC codes and uses these results to optimize code rate profiles via simulated annealing.

Findings

Approximate weight distribution matches exact calculations for small codes.

Designed PAC codes outperform existing codes in simulations.

Method facilitates efficient code design with improved performance.

Abstract

Polarization-adjusted convolutional (PAC) codes combine the polar and convolutional transformations to enhance the distance properties of polar codes. They offer a performance very close to the finite length information-theoretic bounds for short block lengths. In this paper, we develop a method of computing the weight distribution of PAC codes in an approximate form by employing a probabilistic technique. We demonstrate that the results well match the exact weight distributions for small codes that can be computed using a brute-force algorithm. We also present a way employing the results (along with the union bound on the code performance) to design specific PAC codes, more precisely, to determine suitable rate profiles via simulated annealing. Numerical examples illustrate that the PAC codes with the designed rate profiles offer superior performance.