Barker Sequences and Flat Polynomials

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 Barker sequence remains within a constant factor of its L2 norm over the unit circle, in connection with a problem of Littlewood. Second, we show that a Barker sequence produces a polynomial with very large Mahler’s measure, in connection with a question of Mahler. Third, we optimize an argument of Newman to prove that any polynomial with ±1 coefficients and positive degree n−1 has L1 norm less than √ n − .09, and note that a slightly stronger statement would imply that long Barker sequences do not exist. We also record polynomials with ±1 coefficients having maximal L1 norm or maximal Mahler’s measure for each fixed degree up to 24. Finally, we show that if one could establish that the polynomials in a particular sequence are all irreducible over Q, then an alternative proof that there are no long Barker sequences with odd length would follow.