Further Results on Bent Partitions

Bent partitions ofV(p)nplay an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, whereV(p)ndenotes ann-dimensional vector space over the finite field Fp,nis an even positive integer, and p is a prime. It is a challenging open problem whether the depth of any bent partition ofV(p)nis always a power ofp. Notably, the depths of all currently known bent partitions ofV(p)nare powers ofp. In this paper, we prove that for a bent partition Γ ofV(p)nfor which all thep-ary bent functions generated by Γ are regular or all are weakly regular but not regular, the depth of Γ must be a power ofp. We present new constructions of bent partitions that (do not) correspond to vectorial dual-bent functions. In particular, a new construction of vectorial dual-bent functions is provided. Additionally, for general bent partitions ofV(2)n, we establish a characterization in terms of Hadamard matrices.