On the automorphism groups of Lunardon-Polverino scattered linear sets

TL;DR

This paper investigates the automorphism groups and equivalence classes of Lunardon-Polverino scattered linear sets, providing bounds and asymptotic behavior to classify their diversity in projective geometry.

Contribution

It determines the automorphism groups, solves the equivalence problem, and estimates the number of inequivalent Lunardon-Polverino linear sets.

Findings

Automorphism groups explicitly characterized

Equivalence classes among linear sets classified

Bounds and asymptotics for the number of inequivalent sets established

Abstract

Lunardon and Polverino introduced in 2001 a new family of maximum scattered linear sets in $\mathrm{PG}(1,q^n)$ to construct linear minimal R\'edei blocking sets. This family has been extended first by Lavrauw, Marino, Trombetti and Polverino in 2015 and then by Sheekey in 2016 in two different contexts (semifields and rank metric codes). These linear sets are called Lunardon-Polverino linear sets and this paper aims to determine their automorphism groups, to solve the equivalence issue among Lunardon-Polverino linear sets and to establish the number of inequivalent linear sets of this family. We then elaborate on this number, providing explicit bounds and determining its asymptotics.