Rudin-Shapiro-like polynomials in

We examine sequences of polynomials with coefficients constructed using the iterations , where is the degree of and is the reciprocal polynomial of . If these generate the Rudin-Shapiro polynomials. We show that the norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic norm (or, equivalently, with largest possible asymptotic merit factor). The Rudin-Shapiro polynomials form one such sequence.

We determine all 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 of degree 19.