Further results on bent partitions

Bent partitions of $V_{n}^{(p)}$ play an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, where $V_{n}^{(p)}$ denotes an $n$-dimensional vector space over the finite field $\mathbb{F}_{p}$, $n$ 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 $V_{n}^{(p)}$ is always a power of $p$. Notably, the depths of all current known bent partitions of $V_{n}^{(p)}$ are powers of $p$. In this paper, we prove that for a bent partition $\Gamma$ of $V_{n}^{(p)}$ for which all the $p$-ary bent functions generated by $\Gamma$ are regular or all are weakly regular but not regular, the depth of $\Gamma$ 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 $V_{n}^{(2)}$, we establish a characterization in terms of Hadamard matrices.