Algebraic Properties of PAC Codes

TL;DR

This paper investigates the algebraic structure of PAC codes and related codes, revealing properties like duality and minimum distance, and defining a broad class called generalized polynomial polar codes.

Contribution

It introduces generalized polynomial polar codes, including PAC and Reverse PAC codes, and derives their structural properties and limits.

Findings

Structural properties such as duality and minimum distance are established.

Limits on the number of minimum weight codewords and code dimension are deduced.

A broad class of codes encompassing PAC and Reverse PAC codes is defined.

Abstract

We analyze polarization-adjusted convolutional codes using the algebraic representation of polar and Reed-Muller codes. We define a large class of codes, called generalized polynomial polar codes which include PAC codes and Reverse PAC codes. We derive structural properties of generalized polynomial polar codes, such as duality, minimum distance. We also deduce some structural limits in terms of number of minimum weight codewords, and dimension of monomial sub-code.