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Saturating linear sets in PG$(2,q^4)$

Ferdinando Zullo

Abstract

Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study \emph{$1$-saturating} linear sets in PG$(2,q^4)$, that is $\mathbb{F}_q$-linear sets in PG$(2,q^4)$ with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least $5$. This answers to a recent question posed by Bartoli, Borello and Marino.

Saturating linear sets in PG$(2,q^4)$

Abstract

Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study \emph{-saturating} linear sets in PG, that is -linear sets in PG with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least . This answers to a recent question posed by Bartoli, Borello and Marino.
Paper Structure (6 sections, 12 theorems, 28 equations)

This paper contains 6 sections, 12 theorems, 28 equations.

Table of Contents

  1. Preliminaries
  2. Linear sets
  3. Rank-metric codes
  4. Linearized polynomials
  5. Generalized Gabidulin codes
  6. Saturating linear sets in PG$(2,q^4)$ of minimal rank

Key Result

Theorem 1.2

If $L_U$ is an ${\mathbb F}_q$-linear set of rank $n$, with $1 < n \leq m$ on $\mathop{\mathrm{PG}}\nolimits(1,q^m)$, and $L_U$ contains at least one point of weight $1$, then $|L_U| \geq q^{n-1} + 1$.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2: DeBeuleVdV and BoPol
  • Lemma 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • ...and 12 more