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We examine sequences of polynomials withf+1; 1g coecients constructed using the iterations p(x) ! p(x) xd+1p( x), where d is the degree ofp andp is the reciprocal polynomial ofp .I fp0 = 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 Rudin- Shapiro 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 rst really distinct example has a p0 of degree 19.

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