Low-degree planar monomials in characteristic two

TL;DR

This paper characterizes all planar functions of a specific form over finite fields of characteristic two, settling a conjecture and expanding understanding of combinatorial structures in even characteristic fields.

Contribution

It fully determines all planar functions of the form c→uc^t over fields of characteristic two within a certain exponent range, confirming a conjecture by Schmidt and Zhou.

Findings

All such planar functions are classified within the specified parameters.

The results sharpen previous conjectures and provide a complete characterization.

Applications to finite projective planes and combinatorial designs are implied.

Abstract

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications. In this paper we determine all planar functions on F_q of the form c-->uc^t, where q is a power of 2, t is an integer with 0<t<=q^{1/4}, and u is a nonzero element of F_q. This settles and sharpens a conjecture of Schmidt and Zhou.