Further results on bent partitions

Bent partitions of V₍^ (p) play an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, where V₍^ (p) denotes an n-dimensional vector space over the finite field F₏, n is an even positive integer, and p is a prime. For bent partitions, there remains a challenging open problem: Whether the depth of any bent partition of V₍^ (p) is always a power of p. Notably, the depths of all current known bent partitions of V₍^ (p) are powers of p. In this paper, we prove that for a bent partition Γ of V₍^ (p) for which all the p-ary bent functions generated by Γ are regular or all are weakly regular but not regular, the depth of Γ must be a power of p. 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 of V₍^ (2), we establish a characterization in terms of Hadamard matrices.