Planar functions over fields of characteristic two

Abstract

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over ${\mathbb{Z}}_4$. We then specialise to planar monomial functions $f(x)=c{x}^t$ and present constructions and partial results towards their classification. In particular, we show that $t=1$ is the only odd exponent for which $f(x)=c{x}^t$ is planar (for some nonzero $c$) over infinitely many fields. The proof techniques involve methods from algebraic geometry.

23 pages, minor corrections and simplifications compared to the first version

References in corpus (5)

• Planar functions and perfect nonlinear monomials over finite fields
• On the Classification of Exceptional Planar Functions over ${\mathbb{F}}_p$
• Low-degree planar monomials in characteristic two
• New pseudo-planar binomials in characteristic two and related schemes
• (2^n,2^n,2^n,1)-relative difference sets and their representations

Cited by in corpus (5)

• Semifields, relative difference sets, and bent functions
• Rational points on some Fermat curves and surfaces over finite fields
• New pseudo-planar binomials in characteristic two and related schemes
• Exceptional planar polynomials
• On homogeneous planar functions