Further results on bent partitions

Bent partitions of Vₙ⁽ᵖ⁾ play an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, where Vₙ⁽ᵖ⁾ denotes an n-dimensional vector space over the finite field Fₚ, 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 Vₙ⁽ᵖ⁾ is always a power of p. Notably, the depths of all current known bent partitions of Vₙ⁽ᵖ⁾ are powers of p. In this paper, we prove that for a bent partition $Γ$ of Vₙ⁽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 Vₙ⁽²⁾, we establish a characterization in terms of Hadamard matrices.