We present secondary constructions of vectorial functions respectively partitions of elementary abelian groups, which simultaneously yield vectorial dual-bent functions with certain properties, bent partitions, and under some conditions, Latin square partial difference set packings (LP-packings). First, we analyse constructions via the direct sum of vectorial functions and then present a version of the generalized Maiorana-McFarland construction. Next, we generalize a construction of vectorial dual-bent functions by Wang, Fu, and Wei (2023). Finally, we use a lifting procedure of LP-packings from Jedwab and Li (2021) to construct vectorial dual-bent functions, bent partitions, and LP-packings in elementary abelian groups. With these constructions, a large variety of vectorial bent functions, bent partitions, LP-packings, and related amorphic association schemes can be obtained.

Recently, the concept of (normal) bent partitions, which are partitions of elementary abelian groups having similar properties to spreads, has been introduced by Anbar and Meidl. A large number of bent partitions, so-called generalized semifield spreads, can be obtained from semifields with certain properties. A strongly related concept, namely Latin square partial difference set packings (LP-packings) in finite abelian groups, has also been introduced recently by Jedwab and Li. The examples for LP-packings in an elementary abelian group are obtained from spreads. LP-packings yield bent partitions (not only for elementary abelian groups). In this paper, we first point out that conversely, generalized semifield spreads yield LP-packings. As a result, there is a huge amount of LP-packings in elementary abelian groups, other than spreads. With some examples from ternary bent functions, we then show that normal bent partitions and LP-packings are not the same concept. Finally, we extend the lifting procedure from spreads to LP-packings in nonelementary abelian groups to a lifting procedure from some generalized spreads to LP-packings in nonelementary abelian groups and in larger elementary abelian groups. This potentially yields bent partitions other than generalized semifield spreads.