Calabi-Yau complete intersections in fake weighted projective spaces

We present a classification algorithm for Calabi-Yau complete intersections arising from nef-partitions in fake weighted projective spaces, allowing us to determine all such complete intersections up to dimension five. Furthermore, we compute the Hodge pairs of the $3$-dimensional families obtained, and find twenty new Hodge pairs not realized by any toric Calabi-Yau hypersurface. Finally, we provide an explicit characterization for the families of maximal codimension.