  Given a two-level regular fractional factorial design of resolution IV, the
method of doubling produces another design of resolution IV which doubles both
the run size and the number of factors of the initial design. On the other
hand, the projection of a design of resolution IV onto a subset of factors is
of resolution IV or higher. Recent work in the literature of projective
geometry essentially determines the structures of all regular designs of
resolution IV with $n\geq N/4+1$ in terms of doubling and projection, where $N$
is the run size and $n$ is the number of factors. These results imply that, for
instance, all regular designs of resolution IV with $5N/16<n\leq N/2$ must be
projections of the regular design of resolution IV with $N/2$ factors. We show
that, for $9N/32\leq n\leq 5N/16$, all minimum aberration designs are
projections of the design with $5N/16$ factors which is constructed by
repeatedly doubling the 2$^{5-1}$ design defined by $I=\mathit{ABCDE}$. To
prove this result, we also derive some properties of doubling, including an
identity that relates the wordlength pattern of a design to that of its double
and a result that does the same for the alias patterns of two-factor
interactions.
