# Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$

## TL;DR

This paper studies the geometric and algebraic properties of maximum scattered linear sets in projective lines over finite fields, providing new examples and characterizations that relate to MRD-codes.

## Contribution

It characterizes maximum scattered linear sets via vertex properties and constructs new examples, linking geometric configurations to coding theory.

## Key findings

- Characterization of maximum scattered linear sets by vertex properties.
- Construction of new scattered linear sets in PG(1,q^6).
- New MRD-code examples with specific algebraic properties.

## Abstract

In this paper we investigate the geometric properties of the configuration consisting of a $k$-subspace $\Gamma$ and a canonical subgeometry $\Sigma$ in $\mathrm{PG}(n-1,q^n)$, with $\Gamma\cap\Sigma=\emptyset$. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of $\Sigma$ from the vertex $\Gamma$. In particular we deal with the maximum scattered linear sets of the line $\mathrm{PG}(1,q^n)$ found by Lunardon and Polverino and recently generalized by Sheekey. Our aim is to characterize this family by means of the properties of the vertex of the projection as done by Csajb\'ok and the first author of this paper for linear sets of pseudoregulus type. With reference to such properties, we construct new examples of scattered linear sets in $\mathrm{PG}(1,q^6)$, yielding also to new examples of MRD-codes in $\mathbb F_q^{6\times 6}$ with left idealiser isomorphic to $\mathbb F_{q^6}$.