Characterizations of a Class of Planar Functions over Finite Fields
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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 Tr(ax^q+1)+ℓ(x^2) on an extension of the finite field 𝔽_q^n/𝔽_q. Specifically, we investigate those functions on 𝔽_q^2/𝔽_q and construct several typical kinds of planar functions. We also completely characterize them on 𝔽_q^3/𝔽_q. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.
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