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# Characterizations of a Class of Planar Functions over Finite Fields

Mar 2023

Planar functions, introduced by Dembowski and Ostrom, have attracted muchattention in the last decade. As shown in this paper, we present a new class ofplanar functions of the form $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ on anextension of the finite field $\mathbb F_{q^n}/\mathbb F_q$. Specifically, weinvestigate those functions on $\mathbb F_{q^2}/\mathbb F_q$ and constructseveral typical kinds of planar functions. We also completely characterize themon $\mathbb F_{q^3}/\mathbb F_q$. When the degree of extension is higher, itwill be proved that such planar functions do not exist given certainconditions.

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