On the prewellordering associated to the dircted systems of mice

Working under $AD$, we investigate the length of prewellorderings given by the iterates of ${\text{M}}_{2k+1}$, which is the minimal proper class mouse with $2k+1$ many Woodin cardinals. In particular, we answer some questions from \cite{Hjorth01} (the discussion of the questions appears in the last section of \cite{HjorthD}).