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

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 $\Gamma \cap \Sigma =\varnothing$. 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 Csajbók 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 ${\mathbb{F}}_{q^6}$.

References in corpus (10)

• Generalized Twisted Gabidulin Codes
• Algebraic structures of MRD Codes
• New Criteria for MRD and Gabidulin Codes and some Rank-Metric Code Constructions
• On the Genericity of Maximum Rank Distance and Gabidulin Codes
• A Carlitz type result for linearized polynomials
• Identifiers for MRD-codes
• Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants
• MRD-codes arising from the trinomial $x^q+x^{q^3}+c{x}^{q^5}\in {\mathbb{F}}_{q^6}[x]$
• Fuzzy Authentication using Rank Distance
• Invariants and Inequivalence of Linear Rank-Metric Codes

Cited by in corpus (4)

• Linear sets and MRD-codes arising from a class of scattered linearized polynomials
• MRD codes with maximum idealizers
• A new family of maximum scattered linear sets in $\mathrm{PG}(1,q^6)$
• On a conjecture about maximum scattered subspaces of ${\mathbb{F}}_{q^6}\times {\mathbb{F}}_{q^6}$