A Further Study of Vectorial Dual-Bent Functions

Abstract: 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 $$F: V_{n}^{{(p)}\rightarrow} V_{m}<sup>{(p)}$$, where $2\leq m \leq \frac{n}{2}$, $V_{n}<sup>{(p)}$ denotes an $n$-dimensional vector space over the prime field $\mathbb{F}<em>{p}$. 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 $$F: V</em>{n}^{{(p)}\rightarrow} V_{m}<sup>{(p)}$$ with $F(0)=0, F(x)=F(-x)$ and $2\leq m \leq \frac{n}{2}$, 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.