The Barker-sequence conjecture: no Barker sequence of length greater than 13

Corollary 1.2: for $n>1$, a Barker sequence of length $n$ exists if and only if $n\in\{2,3,4,5,7,11,13\}$. The new input is the even case, via the circulant Hadamard theorem; the odd case is the classical Turyn-Storer theorem. Section 5 records examples at every listed length. Nothing is claimed about nearly-Barker sequences or the asymptotic merit-factor problem.

Corollary 1.2: for $n>1$, a Barker sequence of length $n$ exists if and only if $n\in\{2,3,4,5,7,11,13\}$. The new input is the even case, via the circulant Hadamard theorem; the odd case is the classical Turyn-Storer theorem. Section 5 records examples at every listed length. Nothing is claimed about nearly-Barker sequences or the asymptotic merit-factor problem.