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

Let $\mathbb{K}$ be the Galois field ${\mathbb{F}}_{q^t}$ of order $q^t, q=p^e, p$aprime,$A=$\mathrm{A}\mathrm{u}\mathrm{t}$($\mathbb{K}$)$betheautomorphismgroupof$$\mathbb{K}$$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: $${\nu }_{d,\mathit{\sigma }}:⟨v⟩\in \mathrm{P}\mathrm{G}(n-1,\mathbb{K})⟶⟨v^{{\sigma }_0}\otimes v^{{\sigma }_1}\otimes \cdots \otimes v^{{\sigma }_{d-1}}⟩\in \mathrm{P}\mathrm{G}(n^d-1,\mathbb{K})\mathrm{.}$$ Its image will be called the $(d,\mathit{\sigma })$-$Veronese$ $variety$ ${\mathcal{V}}_{d,\mathit{\sigma }}$. Here, we will show that ${\mathcal{V}}_{d,\mathit{\sigma }}$ 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 ${\mathcal{V}}_{d,\mathit{\sigma }}$ and for some choices of parameters, ${\mathcal{V}}_{p,\mathit{\sigma }}$ is the normal rational curve; for $p=2$, it can be the Segre’s arc of $\mathrm{P}\mathrm{G}(3,q^t)$; for $p=3$ ${\mathcal{V}}_{p,\mathit{\sigma }}$ can be also a $\mathrm{\mid }{\mathcal{V}}_{p,\mathit{\sigma }}\mathrm{\mid }$-track of $\mathrm{P}\mathrm{G}(5,q^t)$. Finally, investigate the link between such points sets and a linear code ${\mathcal{C}}_{d,\mathit{\sigma }}$ that can be associated to the variety, obtaining examples of MDS and almost MDS codes.