Classification of Planar Monomials Over Finite Fields of Small Order

TL;DR

This paper computationally verifies that, for finite fields up to size 2^30, the only planar monomials are the known ones, confirming their uniqueness in small finite fields.

Contribution

It provides the first extensive computational proof that no new planar monomials exist in finite fields of size up to 2^30.

Findings

No new planar monomials found in fields up to 2^30

Confirmed the known classification of planar monomials in small finite fields

Supports the conjecture that only known planar monomials exist in small fields

Abstract

For all finite fields of order up to $2^{30}$, we computationally prove that there are no planar monomials besides the ones already known.