Some planar monomials in characteristic 2

Abstract

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They were originally defined only in odd characteristic, but recently Zhou introduced a definition in even characteristic which yields similar applications. In this paper we show that certain functions over ${\mathbb{F}}_{2^r}$ are planar, which proves a conjecture of Schmidt and Zhou. The key to our proof is a new result about the ${\mathbb{F}}_{q^3}$-rational points on the degree-$(q-1)$ Fermat curve $x^{q-1}+y^{q-1}=z^{q-1}$.

7 pages

References in corpus (1)

• Rational points on some Fermat curves and surfaces over finite fields

Cited by in corpus (1)

• Low-degree planar monomials in characteristic two