This work concerns the construction
of Calabi-Yau threefolds in codimension 4. Based on a study of
Hilbert series, we give a list of
families of Calabi-Yau threefolds which may exist in codimension 3 and
codimension 4. Using birational methods, we construct Calabi-Yau threefolds that realize several of the listed families. The main result
is that the cases we consider in
codimension 4 lie in two different deformation components
