Search for Ultraflat Polynomials with Plus and Minus One Coefficients

It is not known whether there exist polynomials with plus and minus one coefficients that are almost constant on the unit circle (called ultraflat). Extensive computations described in this chapter strongly suggest such polynomials do not exist, and lead to conjectures about the precise degree to which flatness can be approached according to various criteria. The evidence shows surprisingly rapid convergence to limiting behavior. Connections to problems about the Golay merit factor, Barker sequences, Golay-Rudin-Shapiro polynomials, and others are discussed. Some results are presented on extensions where the coefficients are allowed to be roots of unity of orders larger than 2. It is pointed out that one conjecture of Littlewood about polynomials with plus and minus one coefficients is true, while another is very likely to be false, as it is inconsistent with another Littlewood conjecture that is supported by the data.