Algebraic Properties of Polar Codes From a New Polynomial Formalism

TL;DR

This paper introduces a new polynomial formalism to analyze the algebraic structure of polar codes, revealing universal properties and symmetries that improve understanding of their minimum weight codewords and decoding performance.

Contribution

It establishes a partial order on monomials generating polar codes, uncovering their algebraic and group-theoretic properties, which were previously not well understood.

Findings

Monomials generating polar codes form a decreasing set under a defined partial order.

The permutation group of these codes contains a large lower triangular affine group.

Minimum weight codewords correspond to orbits of monomials in the generating set.

Abstract

Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attain many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. However, finding the right set of generating monomials for a polar code which optimises the decoding performances is a hard task and channel dependent. The purpose of this paper is to reveal some universal properties of these monomials. We will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised fo a binary-input symmetric channel always form a decreasing set. This property turns out to have rather deep consequences on the structure of the polar code. Indeed, the permutation group of a decreasing monomial code contains a large group called lower triangular affine group. Furthermore, the codewords of minimum weight correspond exactly to the orbits of the minimum weight codewords that are obtained from (evaluations) of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum weight codewords of a decreasing monomial code and henceforth of a polar code.