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：平面函数是由Dembowski和Ostrom引入的，在上个十年中受到了很多关注。 如本文所示，我们在扩展的有限域$ \mathbb F_{q^n}/\mathbb F_q $上提出了一个新的平面函数类$ \operatorname{Tr}(ax^{q + 1})+\ell(x^2)$。具体而言，我们研究了$ \mathbb F_{q^2}/\mathbb F_q $上的这些函数，并构造了几种典型的平面函数。我们还完全描述了它们在$ \mathbb F_{q^3}/\mathbb F_q $上的情况。当扩展的次数更高时，将证明在满足某些条件的情况下，不会存在这种平面函数。
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