$(d,\textbf{\sigma})$-Veronese variety and some applications

TL;DR

This paper introduces a generalized Veronese variety over Galois fields, explores its geometric properties, and links it to linear codes, including MDS and almost MDS codes, revealing new algebraic and combinatorial structures.

Contribution

It defines the $(d,oldsymbol{\sigma})$-Veronese variety, characterizes its linear dependencies, and connects it to optimal linear codes, extending classical algebraic geometry and coding theory.

Findings

The variety is the Grassmann embedding of a rational scroll.

Any $d+1$ points are linearly independent.

Connections to MDS and almost MDS codes are established.

Abstract

Let $\mathbb{K}$ be the Galois field $\mathbb{F}_{q^t}$ of order $q^t, q=p^e, p$ a prime, $A=\mathrm{Aut}(\mathbb{K})$ be the automorphism group of $\mathbb{K}$ and $\boldsymbol{\sigma}=(\sigma_0,\ldots, \sigma_{d-1}) \in A^d$, $d \geq 1$. In this paper the following generalization of the Veronese map is studied: $$ \nu_{d,\boldsymbol{\sigma}} : \langle v\rangle \in \mathrm{PG}(n-1,\mathbb{K}) \longrightarrow \langle v^{\sigma_0} \otimes v^{\sigma_1} \otimes \cdots \otimes v^{\sigma_{d-1}}\rangle \in \mathrm{PG} (n^d-1,\mathbb{K} ). $$ Its image will be called the $(d,\boldsymbol{\sigma})$-$Veronese$ $variety$ $\mathcal{V}_{d,\boldsymbol{\sigma}}$. Here, we will show that $\mathcal{V}_{d,\boldsymbol{\sigma}}$ 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 $\mathcal{V}_{d,\boldsymbol{\sigma}}$ and for some choices of parameters, $\mathcal{V}_{p,\boldsymbol{\sigma}}$ is the normal rational curve; for $p=2$, it can be the Segre's arc of $\mathrm{PG}(3,q^t)$; for $p=3$ $\mathcal{V}_{p,\boldsymbol{\sigma}}$ can be also a $|\mathcal{V}_{p,\boldsymbol{\sigma}}|$-track of $\mathrm{PG}(5,q^t)$. Finally, investigate the link between such points sets and a linear code $\mathcal{C}_{d,\boldsymbol{\sigma}}$ that can be associated to the variety, obtaining examples of MDS and almost MDS codes.