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 $F:V_n^{(p)}\to V_m^{(p)}$, where $2\le m\le \frac{n}{2}$, $V_n^{(p)}$ denotes an $n$-dimensional vector space over the prime field ${\mathbb{F}}_p$. We give new characterizations of certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A) 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. Furthermore, we characterize certain bent partitions in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. For general vectorial dual-bent functions $F:V_n^{(p)}\to V_m^{(p)}$ with $F(0)=0,F(x)=F(-x)$ and $2\le m\le \frac{n}{2}$, we give a necessary and sufficient condition on constructing association schemes. Based on such a result, more association schemes are constructed from vectorial dual-bent functions.