The bent and hyper-bent properties of a class of Boolean functions

This paper considers the bent and hyper-bent properties of a class of Boolean functions of the form $f^{(r)}_{a,b}:=\mathrm{Tr}_{1}^{n}(ax^{r(2^m-1)}) +\mathrm{Tr}_{1}^{4}(bx^{\frac{2^n-1}{5}})$, where $n=2m$, $m\equiv 2k\pmod 4$, $k\in\{0,1\}$, $a\in \mathbb{F}_{2^n}$ and $b\in\mathbb{F}_{16}$. When $m\equiv 2\pmod 4$, we present a detailed description for $f^{(r)}_{a,b}$ to be hyper-bent functions, and give a necessary condition for $f^{(r)}_{a,b}$ to be bent functions when $m\equiv 0\pmod 4$