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 \(\varvec{\sigma }=(\sigma _0,\ldots , \sigma _{d-1}) \in A^d\), \(d \ge 1\). In this paper the following generalization of the Veronese map is studied:

Its image will be called the \((d,\varvec{\sigma })\)-Veronese variety \(\mathcal V_{d,\varvec{\sigma }}\). For \(d=t\), \(\sigma \) a generator of \(\textrm{Gal}({\mathbb F}_{q^t}|{\mathbb F}_{q})\) and \(\varvec{\sigma }=(1,\sigma ,\sigma ^2,\ldots ,\sigma ^{t-1})\), the \((t,\varvec{\sigma })\)-Veronese variety \(\mathcal V_{t,\varvec{\sigma }}\) is the variety studied in [9, 11, 13]. Such a variety is the Grassmann embedding of the Desarguesian spread of \({{\,\mathrm{{PG}}\,}}(nt-1,{\mathbb F}_q)\) and it has been used to construct codes [3] and (partial) ovoids of quadrics, see [9, 12]. Here, we will show that \(\mathcal V_{d,\varvec{\sigma }}\) 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 \(\mathcal V_{d,\varvec{\sigma }}\) and for some choices of parameters, \(\mathcal V_{p,\varvec{\sigma }}\) is the normal rational curve; for \(p=2\), it can be the Segre’s arc of \({{\,\mathrm{{PG}}\,}}(3,q^t)\); for \(p=3\) \(\mathcal V_{p,\varvec{\sigma }}\) can be also a \(|\mathcal V_{p,\varvec{\sigma }}|\)-track of \({{\,\mathrm{{PG}}\,}}(5,q^t)\). Finally, investigate the link between such points sets and a linear code \({\mathcal C}_{d,\varvec{\sigma }}\) that can be associated to the variety, obtaining examples of MDS and almost MDS codes.

设\({\mathbb K}\ )为阶\(q^t, q=p^e, p\)的伽罗华域\({\mathbb F}_{q^t}\ )一个素数，\ (A={{\,\mathrm{{Aut}}\,}}({\mathbb K})\)是\({\mathbb K}\)和\(\varvec{\sigma }的自同构群=(\sigma _0,\ldots , \sigma _{d-1}) \in A^d\) , \(d \ge 1\)。在本文中，研究了 Veronese 映射的以下推广：

它的形象将被称为\((d,\varvec{\sigma })\) - Veronese 品种 \(\mathcal V_{d,\varvec{\sigma }}\)。对于\(d=t\)，\(\sigma \)生成器\(\textrm{Gal}({\mathbb F}_{q^t}|{\mathbb F}_{q})\)和\(\varvec{\sigma }=(1,\sigma ,\sigma ^2,\ldots ,\sigma ^{t-1})\) , \((t,\varvec{\sigma })\ ) -Veronese 变种\(\mathcal V_{t,\varvec{\sigma }}\)是 [9, 11, 13] 中研究的变种。这种多样性是\({{\,\mathrm{{PG}}\,}}(nt-1,{\mathbb F}_q)\)的 Desarguesian 展开的 Grassmann 嵌入它已被用于构造代码 [3] 和二次曲面的（部分）卵形体，请参见 [9、12]。在这里，我们将证明\(\mathcal V_{d,\varvec{\sigma }}\)是正常有理滚动的 Grassmann 嵌入，它的任何\(d+1\)点都是线性独立的。我们给出了\(\mathcal V_{d,\varvec{\sigma }}\)的\(d+2\)个线性相关点的特征，对于一些参数选择，\(\mathcal V_{p,\varvec {\sigma }}\)是正态有理曲线；对于\(p=2\)，它可以是\({{\,\mathrm{{PG}}\,}}(3,q^t)\)的塞格雷弧；对于\(p=3\) \(\mathcal V_{p,\varvec{\sigma }}\)也可以是 \(|\mathcal V_{p,\varvec{\sigma }}|\) - \({{\,\mathrm{{PG}}\,}}(5,q^t)\)的轨迹。最后，调查此类点集与可与品种关联的线性代码\({\mathcal C}_{d,\varvec{\sigma }}\)之间的联系，获得 MDS 和几乎 MDS 代码的示例。