Rudin-Shapiro-like polynomials in L4

We examine sequences of polynomials with {+1,−1} coefficients constructed using the iterations p(x) → p(x) ± xd+1p∗(−x), where d is the degree of p and p∗ is the reciprocal polynomial of p. If p0 = 1 these generate the Rudin-Shapiro polynomials. We show that the L4 norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic L4 norm (or, equivalently, with largest possible asymptotic merit factor). The RudinShapiro polynomials form one such sequence. We determine all p0 of degree less than 40 that generate sequences under the iteration with this property. These sequences have asymptotic merit factor 3. The first really distinct example has a p0 of degree 19.