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A character theoretic approach to planar functions. (English) Zbl 1282.12004

Summary: A function \(f:\mathbb F_p^n\to\mathbb F_p^n\) is called planar if \(x\rightarrowtail f(x+a)-f(x)\) is a permutation for all \(a\neq 0\). In this note we characterize planar functions within a class of functions \(\mathbb F_p^{2m}\to\mathbb F_p^{2m}\) via the planarity of functions \(\mathbb F_p^{2m}\to\mathbb F_p^{2m}\). This class contains some interesting families of planar functions. The proof uses character theoretic arguments.

MSC:

12K10 Semifields
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
11T06 Polynomials over finite fields
51E99 Finite geometry and special incidence structures
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