  Chen and Cheng [Ann. Statist. 34 (2006) 546--558] discussed the method of
doubling for constructing two-level fractional factorial designs. They showed
that for $9N/32\le n\le 5N/16$, all minimum aberration designs with $N$ runs
and $n$ factors are projections of the maximal design with $5N/16$ factors
which is constructed by repeatedly doubling the $2^{5-1}$ design defined by
$I=ABCDE$. This paper develops a general complementary design theory for
doubling. For any design obtained by repeated doubling, general identities are
established to link the wordlength patterns of each pair of complementary
projection designs. A rule is developed for choosing minimum aberration
projection designs from the maximal design with $5N/16$ factors. It is further
shown that for $17N/64\le n\le 5N/16$, all minimum aberration designs with $N$
runs and $n$ factors are projections of the maximal design with $N$ runs and
$5N/16$ factors.
