  Quaternary code (QC) designs form an attractive class of nonregular factorial
fractions. We develop a complementary set theory for characterizing optimal QC
designs that are highly fractionated in the sense of accommodating a large
number of factors. This is in contrast to existing theoretical results which
work only for a relatively small number of factors. While the use of imaginary
numbers to represent the Gray map associated with QC designs facilitates the
derivation, establishing a link with foldovers of regular fractions helps in
presenting our results in a neat form.
