Barker sequences of odd length

TL;DR

This paper presents a new, simpler proof confirming that Barker sequences of odd length greater than 13 do not exist, supporting Turyn's longstanding conjecture.

Contribution

The authors provide a more straightforward proof of Turyn's conjecture regarding the nonexistence of odd-length Barker sequences longer than 13.

Findings

No odd-length Barker sequences longer than 13 exist

The proof simplifies previous complex arguments

Supports Turyn's conjecture with a new approach

Abstract

A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and Storer gave an elementary, though somewhat complicated, proof that this conjecture holds for odd lengths. We give a new and simpler proof of this result.