A note on Barker polynomials

TL;DR

This paper provides a new proof demonstrating the non-existence of Barker polynomials of even degree greater than 12, linking the problem to irreducibility issues and introducing a novel result.

Contribution

It offers a new proof for the non-existence of certain Barker polynomials, connecting the problem to irreducibility questions and presenting a new theoretical result.

Findings

Barker polynomials of even degree > 12 do not exist

Barker sequences of odd length > 13 do not exist

New link between Barker polynomials and irreducibility

Abstract

We give a new proof of the fact that Barker polynomials of even degree greater than 12, and hence Barker sequences of odd length greater than 13 do not exist. This is intimately tied to irreducibility questions and proved as a consequence of a new result.