Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ on an extension of the finite field $\mathbb F_{q^n}/\mathbb F_q$. Specifically, we investigate those functions on $\mathbb F_{q^2}/\mathbb F_q$ and construct several typical kinds of planar functions. We also completely characterize them on $\mathbb F_{q^3}/\mathbb F_q$. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.

翻译：平面函数由Dembowski和Ostrom引入，在过去十年中引起了广泛关注。本文在有限域$\mathbb F_{q^n}/\mathbb F_q$的扩域上，提出了一类形如$\operatorname{Tr}(ax^{q+1})+\ell(x^2)$的新的平面函数。具体而言，我们研究了$\mathbb F_{q^2}/\mathbb F_q$上的此类函数，并构造了若干典型类型的平面函数。同时，我们在$\mathbb F_{q^3}/\mathbb F_q$上对这些函数进行了完全刻画。对于更高次数的扩域，我们证明了在一定条件下这类平面函数不存在。