Characterizations of a Class of Planar Functions over Finite Fields

Ruikai ChenSihem Mesnager

Mar 2023

摘要

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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