Abstract. A Barker sequence is a finite sequence of integers, each ±1, whose aperiodic autocorrelations are all as small as possible. It is widely conjectured that only finitely many Barker sequences exist. We describe connections between Barker sequences and several problems in analysis regarding the existence of polynomials with ±1 coefficients that remain flat over the unit circle according to some criterion. First, we amend an argument of Saffari to show that a polynomial constructed from a… CONTINUE READING