Rudin-Shapiro-like polynomials in $L_{4}$

Abstract:

We examine sequences of polynomials with $\{+1,-1\}$ coefficients constructed using the iterations $p(x)\rightarrow p(x)\pm x^{d+ 1}p^{*}(-x)$, where $d$ is the degree of $p$ and $p^{*}$ is the reciprocal polynomial of $p$. If $p_{0}=1$ these generate the Rudin-Shapiro polynomials. We show that the $L_{4}$ norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic $L_{4}$ norm (or, equivalently, with largest possible asymptotic merit factor). The Rudin-Shapiro polynomials form one such sequence. We determine all $p_{0}$ 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 $p_{0}$ of degree 19.