The norm ratio of the polynomials with coefficients as binary sequence

Given a positive integerq, the ratio of the 2q-norm of a polynomial which its coefficients form a binary sequence and its 2-norm arose from telecommunication engineering consists of finding any type of such polynomials having the ratio “small”. In this paper we consider some special types of these polynomials, discuss the sharpest possible upper bound, and prove a result for the ratio. MAIN FACTS: A conjecture over a Rudin-Shapiro polynomialP _n which has degree 2 ⁿ ?1 is that for any integerq, the ratio of its 2 _q norm and its 2 norm is asymptotic to the 2qth root of 2 ^q (q+1)^?1. In other words $||P_n ||_{2q} \sim ||P_n ||_2 \sqrt{2q}]{{\frac{{2q}}{{q + 1}}}}$ . So far only up toq= 2 has been verified. However if the asymptotic behavior is valid for an evenq, then it is also valid for its next consecutive odd integer.