Irreducible Subcube Partitions

A subcube partition is a of the Boolean cube $\{0,1\}^n$ into subcubes. irreducible if only sub-partitions whose union are singletons and entire partition. tight it “mentions” all coordinates.
 We study extremal properties partitions: minimal size, weight, maximal number points, minimum dimension. also consider existence homogeneous partitions, in which subcubes have same dimensions. additionally partitions $\{0,\dots,q-1\}^n$, $\mathbb{F}_2^n$ affine subspaces, both cases focusing on size.
 Our constructions computer experiments lead to several conjectures values aforementioned properties.