On a class of bent, near-bent, and 2-plateaued functions over finite fields of odd characteristic

The main purpose of this paper is to study a class of the $p$-ary functions $f_{\lambda ,u,v}(x)=T{r}_1^k(\lambda x^{p^k+1})+T{r}_1^n(ux)T{r}_1^n(vx)$ for any odd prime $p$ and $n=2k,\lambda \in GF(p^k{)}^{\ast },u,v\in GF(p^n{)}^{\ast }.$ With the help of Fourier transforms, we are able to subdivide the class of all $f_{\lambda ,u,v}$ into sublcasses of bent, near-bent and 2-plateaued functions. It is shown that the choice of $\lambda ,u$ and $v$, ensuring that $f$ is bent, 2-plateaued or near-bent, is directly related to finding the subset $A\subset GF(p{)}^3$. The efficient method for defining the set $A\subset GF(p{)}^3$ is described in detail.