Infinite Barker series

We say a polynomial p(z)=anzn+an−1zn−1+⋯+a0 is a Littlewood polynomial if ak=±1 for 0⩽k⩽n. Let p(z)p(1/z)=cnzn+cn−1zn−1+⋯+c−nz−n. It is easy to show that c0=n+1. We say that p(z) is a Barker polynomial if |ck|⩽1 for k≠0. There are only 8 known Barker polynomials (normalized to have an=an−1=1). There are many results known about the existence and non-existence of Barker polynomials for various degrees. This paper deals with the infinite case, when f(z)=±1±z±z2±⋯ is a power series with ±1 coefficients. We give a complete description of all Barker series.