Calabi-Yau complete intersections associated to good pairs of generalized nef partitions

We introduce the notion of good pair of generalized nef partitions to describe Calabi-Yau complete intersections in Q-Fano toric varieties whose equations do not necessarily have maximal Newton polytopes. Moreover, we define a natural duality between them which generalizes Batyrev-Borisov mirror duality and allows to define a generalization of Berglund-Hübsch-Krawitz duality to quasismooth complete intersections.