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Characterizations of a Class of Planar Functions over Finite Fields (2303.09265v2)
Published 16 Mar 2023 in math.NT, cs.IT, and math.IT
Abstract: 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 $\operatorname{Tr}(ax{q+1})+\ell(x2)$ on an extension of the finite field $\mathbb F_{qn}/\mathbb F_q$. Specifically, we investigate those functions on $\mathbb F_{q2}/\mathbb F_q$ and construct several typical kinds of planar functions. We also completely characterize them on $\mathbb F_{q3}/\mathbb F_q$. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.