Let K be the Galois field of order q^t , A = Aut(K) be the automorphism group of K and σ = (σ0, . . . , σd−1) in A^d , d ≥ 1. In this paper the following generalization of the Veronese map is studied: νd,σ : 〈v〉 ∈ PG(n − 1, K) −→ 〈v ^σ0 ⊗ v^ σ1 ⊗ · · · ⊗ v^ σd−1 〉 ∈ PG(n d − 1, K). Its image will be called the (d, σ )-Veronese variety Vd,σ . We will show that Vd,σ is the Grassmann embedding of a normal rational scroll and any d + 1 points of it are linearly independent. We give a characterization of d + 2 linearly dependent points of Vd,σ and for some choices of parameters, Vp,σ is the normal rational curve; for p = 2, it can be the Segre’s arc of PG(3, q^t ); for p = 3 Vp,σ can be also a |Vp,σ |-track of PG(5, q^t ). Finally, investigate the link between such points sets and a linear code Cd,σ that can be associated to the variety, obtaining examples of MDS and almost MDS codes.

(d, σ) -Veronese variety and some applications / Durante, N.; Longobardi, G.; Pepe, V.. - In: DESIGNS, CODES AND CRYPTOGRAPHY. - ISSN 0925-1022. - (2023). [10.1007/s10623-023-01186-9]

(d, σ) -Veronese variety and some applications

Pepe V.
2023

Abstract

Let K be the Galois field of order q^t , A = Aut(K) be the automorphism group of K and σ = (σ0, . . . , σd−1) in A^d , d ≥ 1. In this paper the following generalization of the Veronese map is studied: νd,σ : 〈v〉 ∈ PG(n − 1, K) −→ 〈v ^σ0 ⊗ v^ σ1 ⊗ · · · ⊗ v^ σd−1 〉 ∈ PG(n d − 1, K). Its image will be called the (d, σ )-Veronese variety Vd,σ . We will show that Vd,σ is the Grassmann embedding of a normal rational scroll and any d + 1 points of it are linearly independent. We give a characterization of d + 2 linearly dependent points of Vd,σ and for some choices of parameters, Vp,σ is the normal rational curve; for p = 2, it can be the Segre’s arc of PG(3, q^t ); for p = 3 Vp,σ can be also a |Vp,σ |-track of PG(5, q^t ). Finally, investigate the link between such points sets and a linear code Cd,σ that can be associated to the variety, obtaining examples of MDS and almost MDS codes.
2023
veronese variety; finite fields; field automorphisms, caps; linear codes
01 Pubblicazione su rivista::01a Articolo in rivista
(d, σ) -Veronese variety and some applications / Durante, N.; Longobardi, G.; Pepe, V.. - In: DESIGNS, CODES AND CRYPTOGRAPHY. - ISSN 0925-1022. - (2023). [10.1007/s10623-023-01186-9]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1675032
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