A Further Study of Vectorial Dual-Bent Functions

Vectorial dual-bent functions have recently attracted some researchers’ interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions, and linear codes. In this paper, we further study vectorial dual-bent functions $F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$ , where $2\leq m \leq \frac {n}{2}$ , and $V_{n}^{(p)}$ denotes an n-dimensional vector space over the prime field $\mathbb {F}_{p}$ . For certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A), we present a more concise characterization in terms of partial difference sets than the one given in Wang et al. (2023), and give new characterizations in terms of amorphic association schemes, linear codes, and generalized Hadamard matrices, respectively. When $p=2$ , we characterize vectorial dual-bent functions with Condition A in terms of bent partitions. Through the relationship between vectorial dual-bent functions and bent partitions, new characterizations of certain bent partitions in terms of amorphic association schemes, linear codes, and generalized Hadamard matrices are obtained. For a vectorial dual-bent function $F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$ with $F(0)=0, F(x)=F(-x)$ , where $2\leq m \leq \frac {n}{2}$ , we give a necessary and sufficient condition under which the preimage set partition of F induces an association scheme. By using two classes of vectorial dual-bent functions, more association schemes are obtained.