Maximum scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^4)$

TL;DR

This paper proves that for the projective line over a finite field extension of degree four, the only maximum scattered linear sets are those of pseudoregulus type and the ones discovered by Lunardon and Polverino, classifying their symmetry orbits.

Contribution

It establishes the uniqueness of maximum scattered linear sets in PG(1,q^4) and describes their orbit structure under key projective groups.

Findings

Only two families of maximum scattered linear sets exist in PG(1,q^4).

The orbits of these linear sets under PGL(2,q^4) and PΓL(2,q^4) are fully characterized.

The result confirms the classification for t=4, extending understanding of linear sets.

Abstract

There are two known families of maximum scattered $\mathbb{F}_q$-linear sets in $PG(1,q^t)$: the linear sets of pseudoregulus type and for $t\geq 4$ the scattered linear sets found by Lunardon and Polverino. For $t=4$ we show that these are the only maximum scattered $\mathbb{F}_q$-linear sets and we describe the orbits of these linear sets under the groups $PGL(2,q^4)$ and $P\Gamma L(2,q^4)$.