Characterizations of a Class of Planar Functions over Finite Fields

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 $\mathrm{Tr}(a{x}^{q+1})+\mathrm{\ell }(x^2)$ on an extension of the finite field ${\mathbb{F}}_{q^n}\mathrm{/}{\mathbb{F}}_q$. Specifically, we investigate those functions on ${\mathbb{F}}_{q^2}\mathrm{/}{\mathbb{F}}_q$ and construct several typical kinds of planar functions. We also completely characterize them on ${\mathbb{F}}_{q^3}\mathrm{/}{\mathbb{F}}_q$. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.