The merit factor of binary sequence families constructed from m-sequences

We consider the asymptotic merit factor of two binary sequence families obtained from an initial binary sequence family using a “negaperiodic” and a “periodic” construction. When the initial sequences are m-sequences, both of the constructed families have the same asymptotic merit factor as the initial family, at all rotations of sequence elements. A similar property was previously shown to hold when the initial sequences are Legendre sequences. However we show by example that this property appears to fail for a general initial sequence family.