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) = Tr_1^k(\lambda x^{p^k+1})+Tr^n_1(ux)Tr_1^n(vx) $ for any odd prime $ p $ and $ n = 2k, \lambda\in GF(p^k)^*, u, v\in GF(p^n)^*. $ 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.