Let $\mathbb{K}$ be a Galois field $\mathbb{F}_{q^t}$ of order $q^t, q=p^e, p$ prime, $A=\mathrm{Aut }(\mathbb{K})$ is 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. ) \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} ). $$ The image is called $(d,\boldsymbol{\sigma})$-$Veronese$ $variety$ $\mathcal{V}_{d,\boldsymbol{\sigma}}$. Here we show that $\mathcal{V}_{d,\boldsymbol{\sigma}}$ is the Grassmann embedding of the ordinary rational scroll, whose arbitrary $d+1$ points are linearly independent. increase. We characterize the $d+2$ linearly dependent points of $\mathcal{V}_{d,\boldsymbol{\sigma}}$ and for some choice of parameters, $\mathcal{V}_{ p,\boldsymbol{\sigma}}$ is the usual rational curve. For $p=2$ it becomes a Segre arc of $\mathrm{PG}(3,q^t)$ . If $p=3$ $\mathcal{V}_{p,\boldsymbol{\sigma}}$ is also $|\mathcal{V}_{p,\boldsymbol{\sigma}}|$-track At the end of $\mathrm{PG}(5,q^t)$ we have such a point set and a linear code $\mathcal{C}_{d,\boldsymbol{\sigma} that can be associated with the diversity Investigate the links between }$ and get examples of MDS and almost MDS code.