1 Introduction
Let \(V=V(n,{\mathbb K})\) be an n-dimensional vector space over a field \({\mathbb K}\), we will denote by \({{\,\mathrm{{PG}}\,}}(V)\) as well as \({{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K})\) the projective space induced by it.
We refer to [5] for the definition of dimension, degree, smoothness and tangent space of an algebraic variety and for the methods and techniques used to study classical varieties.
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The Veronese variety \(\mathcal {V}_d\) of degree d and dimension \(n-1\) is a classical algebraic variety widely studied over fields of any characteristic [5, 6] and it is the image of the Veronese mapwhere \(X_I\) ranges over all the possible monomials of degree d in \(x_0,x_1,\ldots ,x_{n-1}\).
$$\begin{aligned} \nu _d: (x_0,x_1,\ldots ,x_{n-1}) \in {{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K}) \longrightarrow (\ldots ,X_I,\ldots ) \in {{\,\mathrm{{PG}}\,}}\left( \left( {\begin{array}{c}n+d-1\\ d\end{array}}\right) -1,{\mathbb K}\right) , \end{aligned}$$
The Veronese map can be defined also byNow, let \(V_i\) be \(n_i\)-dimensional vector spaces over the field \({\mathbb K}\), \(i=0,1,\ldots ,d-1\). A Segre variety of type \((n_0,n_1,\ldots ,n_{d-1})\) in \({{\,\mathrm{{PG}}\,}}\bigl (\bigotimes _{i=0}^{d-1} V_i \bigr )\) is the setIf \(n_0=\cdots =n_{d-1}=n\), we write \(\Sigma _{(n-1)^d}\) instead of \(\Sigma _{n-1,n-1,\ldots ,n-1}\). Then it is clear that \(\mathcal V_{d}\) turns out to be a linear section of the Segre variety product of \({{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K})\) for itself d times.
$$\begin{aligned} \nu _{d} : \langle v \rangle \in {{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K}) \longrightarrow \langle v \otimes v \otimes \cdots \otimes v \rangle \in {{\,\mathrm{{PG}}\,}}\left( \left( {\begin{array}{c}n+d-1\\ d\end{array}}\right) -1,{\mathbb K}\right) . \end{aligned}$$
$$\begin{aligned} \Sigma _{n_0-1,n_1-1,\ldots ,n_{d-1}-1}= \bigl \{\langle v_0 \otimes v_1 \otimes \cdots \otimes v_{d-1} \rangle \,\,|\,\, v_i \in V_i \setminus \{0\}, \, i=0,1,\ldots ,d-1 \bigr \}.\nonumber \\ \end{aligned}$$
(1)
If \(\mathbf {\zeta }\) is a collineation of \({{\,\mathrm{{PG}}\,}}\bigl (V^{\otimes d} \bigl )\) fixing \(\Sigma _{(n-1)^d}\), then there exist \(\zeta _i, i=0,1,\ldots ,d-1\) semilinear maps of \({{\,\mathrm{{PG}}\,}}(V)\), with the same companion field automorphism, and a permutation \(\tau \) on \(\{0,1,\ldots ,d-1\}\) such thatfor a proof of this in positive characteristic see [16].
$$\begin{aligned} \langle v_0 \otimes v_1 \otimes \cdots \otimes v_{d-1} \rangle ^{\mathbf {\zeta }}=\left\langle v_{\tau (0)}^{\zeta _0}\otimes v_{\tau (1)}^{\zeta _1}\otimes \cdots \otimes v_{\tau (d-1)}^{\zeta _{d-1}}\right\rangle , \end{aligned}$$
Let \(\mathcal {L}_h\) be the set of all projective subspaces of dimension h of \({{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K})\), and considerwhere \(\wedge \) is the wedge product and \(\bigwedge ^{h+1}V\) the \((h+1)\)th exterior power of V. This map is called Grassmann embedding and its image \(\mathcal {G}_{n,h}(V)\) is called Grassmannian of subspaces of dimension h of \({{\,\mathrm{{PG}}\,}}(V)\). It is well-known that \(\mathcal {G}_{n,h}(V)\) is an algebraic variety which is the complete intersection of certain quadrics, see [5].
$$\begin{aligned} g_{n,h}: \langle v_0, v_1,v_2,\ldots ,v_h \rangle \in \mathcal {L}_h \longrightarrow \langle v_0 \wedge v_1 \wedge v_2 \wedge \cdots \wedge v_h \rangle \in {{\,\mathrm{{PG}}\,}}\bigl ( \bigwedge \,^{h+1}V \bigr ), \end{aligned}$$
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Let \({\mathbb K}\) be the Galois field \({\mathbb F}_{q^t}\) of order \(q^t\), \(A={{\,\mathrm{{Aut}}\,}}({\mathbb K})\) be the automorphism group of \({\mathbb K}\) and \(\varvec{\sigma }=(\sigma _0,\ldots , \sigma _{d-1}) \in A^d\), \(d \ge 1\). The aim of this paper is to study the following generalization of the Veronese mapand some properties of its image that will be called here 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 [4] and (partial) ovoids of quadrics, see [9, 12].
$$\begin{aligned} \nu _{d,\varvec{\sigma }} : \langle v \rangle \in {{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K}) \longrightarrow \langle v^{\sigma _0} \otimes v^{\sigma _1} \otimes \cdots \otimes v^{\sigma _{d-1}} \rangle \in {{\,\mathrm{{PG}}\,}}(n^d-1,{\mathbb K}) \end{aligned}$$
A \([\nu ,\kappa ]\)-linear code \({\mathcal C}\) is a subspace of the vector space \({\mathbb F}_{q}^\nu \) of dimension \(\kappa \). The weight of a codeword is the number of its entries that are nonzero and the Hamming distance between two codewords is the number of entries in which they differ. The distance \(\delta \) of a linear code is the minimum distance between distinct codewords and it is equals to the minimum weight. A linear code of length \(\nu \), dimension \(\kappa \), and minimum distance \(\delta \) is called a \([\nu ,\kappa ,\delta ]\)-code. A matrix H of order \((\nu -\kappa ) \times \nu \) such thatis called a parity check matrix for \({\mathcal C}\). The minimum weight, and hence the minimum distance, of \({\mathcal C}\) is at least w if and only if any \(w-1\) columns of H are linearly independent [10, Theorem 10, p. 33]. Each linear \([\nu ,\kappa ,\delta ]\)-code \({\mathcal C}\) satisfies the following inequalitycalled Singleton bound. If \(\delta =\nu -\kappa +1\), \({\mathcal C}\) is called maximum distance separable or MDS, while if \(\delta =\nu -\kappa \) the code is called almost MDS. These can be related to some subsets of points in the projective space. More precisely, \({\mathcal C}\) is a \([\nu ,\kappa ,\delta ]\)-linear code if and only if the columns of its parity check matrix H can be seen as \(\nu \) points in \({{\,\mathrm{{PG}}\,}}(\nu -\kappa -1,q)\) each \(\delta -1\) of which are in general position, [2, Theorem 1]. Then, the existence of MDS or almost MDS linear codes is equivalent to the existence of arcs or tracks in projective spaces, respectively.
$$\begin{aligned} {\textbf{x}}H^T={\textbf{0}}\quad \text {for all} \,\, {\textbf{x}} \in {\mathcal C}\end{aligned}$$
$$\begin{aligned} \delta \le \nu -\kappa +1, \end{aligned}$$
Definition 1.1
A k-arc is a set of k points in \({{\,\mathrm{{PG}}\,}}(r,q)\) such that \(r+1\) of them are in general position. An \(\ell \)-track is a set of \(\ell \) points in \({{\,\mathrm{{PG}}\,}}(r,q)\) such that every r of them are in general position.
Here, we study the variety \(\mathcal V_{d,\varvec{\sigma }}\) and we will prove that it is the Grassmann embedding of a normal rational scroll and that any \(d+1\) points of it are in general position, i.e. any \(d+1\) points of \(\mathcal V_{d,\varvec{\sigma }}\) are linearly independent. Moreover, we give a characterization of \(d+2\) linearly dependent points of this variety and investigate how such a property is interesting for a linear code \({\mathcal C}_{d,\varvec{\sigma }}\) that can be associated to the variety.
2 The variety \(\mathcal V_{d,\varvec{\sigma }}\)
Let \(V=V(n,{\mathbb K})\) be an n-dimensional vector space over the field \({\mathbb K}\) and \({{\,\mathrm{{PG}}\,}}(V)={{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K})\) be the induced projective space. In particular, if \({\mathbb K}\) is the Galois field of order \(q^t\), we will denote the projective space by \({{\,\mathrm{{PG}}\,}}(n-1,q^t)\)
Let \(A={{\,\mathrm{{Aut}}\,}}({\mathbb K})\) be the automorphism group of \({\mathbb K}\) and \(\varvec{\sigma }=(\sigma _0,\ldots , \sigma _{d-1}) \in A^d\), \(d \ge 1\), and define the mapUp to the action of the group P\(\Gamma \)L(V), we may assume that \(\sigma _0=1\). It is clear that the map \(\nu _{d,\varvec{\sigma }}\) is an injection of \({{\,\mathrm{{PG}}\,}}(V)\) into \({{\,\mathrm{{PG}}\,}}(V^{\otimes d})\) by the injectivity of the Segre map.
$$\begin{aligned} \nu _{d,\varvec{\sigma }} : \langle v \rangle \in {{\,\mathrm{{PG}}\,}}(V) \longrightarrow \langle v^{\sigma _0} \otimes v^{\sigma _1} \otimes \cdots \otimes v^{\sigma _{d-1}} \rangle \in {{\,\mathrm{{PG}}\,}}\bigl (V^{\otimes d} \bigl ). \end{aligned}$$
(2)
We will call \(\nu _{d,\varvec{\sigma }}\) the \((d,\varvec{\sigma })\)-Veronese embedding and, as defined before, its image \(\mathcal V_{d,\varvec{\sigma }}\) the \((d,\varvec{\sigma })\)-Veronese variety. Then \(\mathcal V_{d,\varvec{\sigma }}\) is a rational variety of dimension \(n-1\) in \({{\,\mathrm{{PG}}\,}}(N-1,{\mathbb K})\), \(N=n^d\) and it has as many points as \({{\,\mathrm{{PG}}\,}}(n-1,q^t)\), see [5, 6].
As a consequence of [16, Theorems 3.5 and 3.8], one gets the following
Theorem 2.1
Let \(\zeta \) be a collineation of \({{\,\mathrm{{PG}}\,}}\bigl (V^{\otimes d} \bigl )\). Then \(\zeta \) fixes \(\mathcal V_{d,\varvec{\sigma }}\) if and only ifwhere \(\zeta _0,\) is a bijective semilinear map of V.
$$\begin{aligned} \left\langle v\otimes v^{\sigma _1}\otimes \cdots \otimes v^{\sigma _{d-1}} \right\rangle ^{\mathbf {\zeta }}= \left\langle v^{\zeta _0}\otimes v^{ \zeta _0\sigma _1}\otimes \cdots \otimes v^{ \zeta _{0}\sigma _{d-1}} \right\rangle \quad \text {for any}\,{ v \in V \setminus \{0\}} \end{aligned}$$
Note that applying the mapwhere \(\zeta _i\) is a bijective semilinear map, we get a subvariety of \(\Sigma _{(n-1)^d}\) projectively equivalent to \(\mathcal V_{d,\varvec{\sigma }}\).
$$\begin{aligned} \left\langle v\otimes v^{\sigma _1}\otimes \cdots \otimes v^{\sigma _{d-1}}\right\rangle ^{\mathbf {\zeta } }=\left\langle v^{\zeta _0}\otimes v^{\zeta _1\sigma _1}\otimes \cdots \otimes v^{\zeta _{d-1}\sigma _{d-1}} \right\rangle , \end{aligned}$$
Although many of the results also hold in the case of a general field, from now on it will be assumed, that \({\mathbb K}\) is the Galois field \({\mathbb F}_{q^t}\) of \(q^t\) elements and \(\varvec{\sigma }=(\sigma _0,\sigma _1,\ldots ,\sigma _{d-1}) \in G^d\) with \(G={{\,\textrm{Gal}\,}}({\mathbb F}_{q^t} |\, {\mathbb F}_q)\). Moreover, since any element \(\sigma _i \in G\) is a map of the type \(\sigma _i : x \mapsto x^{q^{h_i}}\) with \(0 \le h_i < t\) and \(0 \le i \le d-1\), hereafter we will suppose thatwhere \(0=h_0<h_1<\cdots<h_m<t\) and we will consider the vector \(d_{\varvec{\sigma }}=(d_0,d_1,\ldots ,d_m)\) where \(d_j\) is the occurrence of \(\sigma _j\) in \(\varvec{\sigma }\), \(0 \le j \le m\). Clearly \(d_0+d_1+\ldots +d_m=d\). If \(\varvec{\sigma }\in G^d\), the integerwill be called norm of \(\varvec{\sigma }\).
$$\begin{aligned} \varvec{\sigma }=(\underbrace{\sigma _0,\ldots ,\sigma _0}_{d_0\, \text {times}},\underbrace{\sigma _1,\ldots ,\sigma _1}_{d_1\,\text {times}}, \ldots ,\underbrace{\sigma _{m},\ldots ,\sigma _{m}}_{d_m\, \text {times}}), \end{aligned}$$
$$\begin{aligned} |\varvec{\sigma }|= \sum _{i=0}^{d-1}q^{h_i}=\sum _{i=0}^{m}d_iq^{h_i}. \end{aligned}$$
(3)
Since we consider the ring of polynomials \({\mathbb F}_{q^t}[x_0,x_1,\ldots ,x_{n-1}]\) actually as the quotient \({\mathbb F}_{q^t}[x_0,x_1,\ldots ,x_{n-1}]/(x_0^{q^t}-x_0,x_1^{q^t}-x_1,\ldots ,x_{n-1}^{q^t}-x_{n-1})\), from now on we assume \(|\varvec{\sigma }|< q^t\), so that distinct polynomials will be distinct functions over \({\mathbb F}_{q^t}\). By injectivity of map in (2), it is clear that \((d,\varvec{\sigma })\)-Veronese variety \(\mathcal V_{d,\varvec{\sigma }}\) has as many points as \({{\,\mathrm{{PG}}\,}}(n-1,q^t)\).
Let \(\{e_i \, \,|\,\,i=0,1,\ldots ,nd-1\}\) be the canonical basis of \(V(nd,{\mathbb F}_{q^t})=V(nd,q^t)\) and let \(\Pi \cong {{\,\mathrm{{PG}}\,}}(n-1,q^t)\) be the subspace of \({{\,\mathrm{{PG}}\,}}(nd-1,q^t)\) spanned by \(\{\langle e_i \rangle \,\,|\,\,0 \le i \le n-1\}\). Let \(\phi \) be the collineation of \({{\,\mathrm{{PG}}\,}}(nd-1,q^t)\) such thatwhere the subscripts are taken modulo nd. As done in [4, Sect. 4], for any \(\langle v_i \rangle \in \Pi ^{\phi ^i}\), we can identify \(v_0\otimes v_1\otimes v_2\otimes \cdots \otimes v_{d-1}\) with \(v_0\wedge v_1\wedge v_{2}\wedge \cdots \wedge v_{d-1}\). Therefore, \(\mathcal V_{d,\varvec{\sigma }}\) is the Grassmann embedding of the d-fold normal rational scrollof \({{\,\mathrm{{PG}}\,}}(nd-1,q^t)\), see [5, Chap. 8] for a definition of normal rational scroll.
$$\begin{aligned} \langle e_i \rangle \mapsto \langle e_{i+n} \rangle , \end{aligned}$$
$$\begin{aligned} S^{\varvec{\sigma }}_{n-1,n-1,\ldots ,n-1}=\bigl \{\left\langle P^{\sigma _0 },P^{\phi \sigma _1},P^{\phi ^2 \sigma _2},\ldots ,P^{\phi ^{d-1} \sigma _{d-1}}\bigr \rangle \, |\,P \in \Pi \right\} \end{aligned}$$
Example 2.2
Let \(\varvec{\sigma }=\varvec{1}\), the identity of the product group \(G^d\), the \((d,\varvec{\sigma })\)-Veronese variety \(\mathcal V_{d,\varvec{\sigma }}\) is the classical Veronese variety of degree d and \(\mathcal V_{d,\varvec{\sigma }} \subset {{\,\mathrm{{PG}}\,}}(N-1,q^t)\) with \(N=\left( {\begin{array}{c}n+d-1\\ d\end{array}}\right) \). In this case, \(\mathcal V_{d,\varvec{\sigma }}\) is the Grassmann embedding of \(S_{n-1,n-1,\ldots ,n-1}=\{\langle P,P^{\phi },P^{\phi ^2},\ldots ,P^{\phi ^{d-1}}\rangle \, |\,P \in \Pi \}\), i.e. the Segre variety \(\Sigma _{n-1,d-1}\) of \({{\,\mathrm{{PG}}\,}}(nd-1,q^t)\), see again [5, Chap. 8].
Example 2.3
Let \(\sigma \) be a generator of \({{\,\textrm{Gal}\,}}({\mathbb F}_{q^t}|{\mathbb F}_q)\) and \(\varvec{\sigma }=(1,\sigma ,\ldots ,\sigma ^{t-1})\), then we get the algebraic variety introduced in [9, 11, 13] and we will refer to it as the SLP-variety \(\mathcal V_{t,\sigma }\). Let \(\hat{\sigma }\) be the semi-linear collineation \(\phi \circ \sigma \) of \({{\,\mathrm{{PG}}\,}}(nt-1,q^t)\) of order t. Then the set of points fixed by \(\hat{\sigma }\), \(\textrm{Fix}(\hat{\sigma })\subset {{\,\mathrm{{PG}}\,}}(nt-1,q^t)\), is a subgeometry isomorphic to \({{\,\mathrm{{PG}}\,}}(nt-1,q)\) and a subspace of \({{\,\mathrm{{PG}}\,}}(nt-1,q^t)\) intersects the subgeometry in a subspace of the same dimension if and only if it is set-wise fixed by \(\hat{\sigma }\) (see [9, Sect. 3]). In this caseand hence its \((t-1)\)-spaces are set-wise fixed by \(\hat{\sigma }\). Also, \(S_{n-1,n-1,\ldots ,n-1}\cap \textrm{Fix}(\hat{\sigma })\) is the Desarguesian \((t-1)\)-spread of \({{\,\mathrm{{PG}}\,}}(nt-1,q)=\textrm{Fix}(\hat{\sigma })\subset {{\,\mathrm{{PG}}\,}}(nt-1,q^t)\). Therefore, \(\mathcal V_{t,\sigma }\) is the Grassmann embedding of the Desarguesian spread of \({{\,\mathrm{{PG}}\,}}(nt-1,q)\). In this case, in fact, \(\mathcal V_{t,\sigma }\) turns out to be a variety of the subgeometry \({{\,\mathrm{{PG}}\,}}(n^t-1,q)\subset {{\,\mathrm{{PG}}\,}}(n^t-1,q^t)\) point-wise fixed by the semi-linear collineation of order t of \({{\,\mathrm{{PG}}\,}}(n^t-1,q^t)\) induced by \(\hat{\sigma }\):
$$\begin{aligned} S^{\varvec{\sigma }}_{n-1,n-1,\ldots ,n-1}=\bigl \{\left\langle P,P^{\hat{\sigma }},P^{\hat{\sigma }^2},\ldots ,P^{\hat{\sigma }^{t-1}}\right\rangle \, |\,P \in \Pi \bigr \}, \end{aligned}$$
$$\begin{aligned} v_0\otimes v_1\otimes \cdots \otimes v_{t-1}\mapsto v_{t-1}^{\hat{\sigma }} \otimes v_0^{\hat{\sigma }}\otimes \cdots \otimes v_{t-2}^{\hat{\sigma }}. \end{aligned}$$
By (2), a point of \({{\,\mathrm{{PG}}\,}}(n-1,q^t)\) with homogeneous coordinates \((x_0,x_1,\ldots ,x_{n-1})\) is mapped by \(\nu _{d,\varvec{\sigma }}\) into a point of coordinateswhere \(X_{I_j}\) is a monomial of degree \(d_j\) in the variables \(x_0,x_1,\ldots ,x_{n-1}\). Hence, the \((d,\varvec{\sigma })\)-Veronese variety \(\mathcal V_{d,\varvec{\sigma }}\) is contained in a projective space of vector space dimensionLet \(\sigma _0,\sigma _1,\ldots \sigma _m\) distinct automorphisms in \(\varvec{\sigma }\) and \(d_{\varvec{\sigma }}\) the vector of their occurrences and suppose that \(d_i\sigma _i\ne d_j\sigma _j\) for all \(i,j=0,1,\ldots ,m\) distinct, then we get exactly \(N=n^d\) distinct monomials of type \(\displaystyle \prod \nolimits _{j=0}^{m}X_{I_j}^{\sigma _j}\). This is not the case anymore if \(d_i\sigma _i=d_j\sigma _j\) for some \(i\ne j\). For example, if \(q=2\), \(\varvec{\sigma }=(1,1,2)\), then \(d_0=2,d_1=1\) and hence \(d_0=d_1\sigma _1\). Thenand we get 5 distinct monomials and \(\mathcal V_{3,\varvec{\sigma }}\) is in fact contained in a projective space of vector space dimension less than \(N=6\).
$$\begin{aligned} \left( \ldots ,\displaystyle \prod _{j=0}^{m}X_{I_j}^{\sigma _j},\ldots \right) , \end{aligned}$$
$$\begin{aligned} N=N_0N_1\cdots N_m, \quad N_j=\left( {\begin{array}{c}n+d_j-1\\ d_j\end{array}}\right) ,\quad j=0,1,\ldots ,m. \end{aligned}$$
(4)
$$\begin{aligned} (x_0,x_1)\otimes (x_0,x_1)\otimes \left( x_0^2,x_1^2\right) =\left( x_0^4,x_0^2x_1^2,x_0^3x_1,x_0x_1^3,x_0^3x_1,x_0x_1^3,x_0^2x_1^2,x_1^4\right) , \end{aligned}$$
Recall that an r-hypersurface of \({{\,\mathrm{{PG}}\,}}(n-1,q^t)\) is a variety such that its points have coordinates vanish an r-form of \({\mathbb F}_{q^t}[X_0,\ldots ,X_{n-1}].\) If \(r=2\), an r-hypersurface is called quadric. In [14], it is shown a lower bound on the degree of an r-hypersurface \(\mathcal {D}\) of \({{\,\mathrm{{PG}}\,}}(n-1,q^t)\) after which \(\mathcal {D}\) could contain all points of the projective space. More precisely,
Theorem 2.4
[14] If an r-hypersurface \(\mathcal {D}\) of \({{\,\mathrm{{PG}}\,}}(n-1,q^t)\) contains all the points of the space, then \(r \ge q^t+1\).
Let I be a multi-index of the form \(I=I_0I_1\cdots I_m\), where \(I_j\) is a multi-index corresponding to a monomial in \(x_0,x_1,\ldots ,x_{n-1}\) of degree \(d_j\). Once we have labelled the coordinates of \({{\,\mathrm{{PG}}\,}}(N-1,q^t)\) according to the multi-index I, we can define a natural linear map \(\psi \) that sends the hyperplane of \({{\,\mathrm{{PG}}\,}}(N-1,q^t)\) of equation \(\displaystyle \sum \nolimits _{I}a_Iz_I=0\) to the \(\varvec{\sigma }\)-hypersurface of equationThen, by Theorem 2.4, we get the following result.
$$\begin{aligned} \sum _{I}a_I \displaystyle \prod _{j=0}^{m}X_{I_j}^{\sigma _j}=0. \end{aligned}$$
Theorem 2.5
Let \(\varvec{\sigma }\in G^d\) with \(d_{\varvec{\sigma }}=(d_0,d_1,\ldots ,d_m),\) \(| \varvec{\sigma }|<q^t\). The \((d,\varvec{\sigma })\)-Veronese variety \(\mathcal V_{d,\varvec{\sigma }}\) is not contained in any hyperplane of \({{\,\mathrm{{PG}}\,}}(N-1,q^t)\) with \(N=N_0 N_1\cdots N_m\) and
$$\begin{aligned}N_j=\left( {\begin{array}{c}n+d_j-1\\ d_j\end{array}}\right) , \quad j=0,1,\ldots ,m. \end{aligned}$$
In the following, we generalize some results proved in [3, Sect. 2] for the SLP-variety.
Theorem 2.6
Let \(\Pi _0,\Pi _1,\ldots ,\Pi _{d-1}\) be proper subspaces of \({{\,\mathrm{{PG}}\,}}(n-1, q^t)\) and suppose that \(P \in {{\,\mathrm{{PG}}\,}}(n-1, q^t)\) is not contained in any of them. Then, \(P^{\nu _{d,\varvec{\sigma }}}\) is not contained in \(\langle \Pi _0^{\nu _{d,\varvec{\sigma }}},\Pi _1^{\nu _{d,\varvec{\sigma }}},\ldots ,\Pi _{d-1}^{\nu _{d,\varvec{\sigma }}} \rangle \).
Proof
Recall that the dual space of \(V(n^d,q^t)\), denoted by \(V(n^d,q^t)^*\), is spanned by the simple tensors \(l_0^ *\otimes l_1^*\otimes \cdots \otimes l_{d-1}^*\), with \(l_i^* \in V(n,q^t)^*\), and \(l_0^*\otimes l_1^*\otimes \cdots \otimes l_{d-1}^*\) evaluated in \(u_0\otimes u_1 \otimes \cdots \otimes u_{d-1}\) is \(l_0^*(u_0)l_1^*(u_1)\cdots l_{d-1}^*(u_{d-1}) \in {\mathbb {F}}_{q^t}\).
For every \(i \in \{0,1,\ldots ,d-1\}\), take an \(l_i^* \in V(n,q^t)^*\) such that \(l_i^*\) vanishes on \(\Pi _i^{\sigma _i}\) and not in \(P^{\sigma _i}\). Then the hyperplane defined by \( l_0^*\otimes l_1^*\otimes \cdots \otimes l_{d-1}^*\) contains the points of \(\Pi _j^{\nu _{d,\varvec{\sigma }}}\) \(\forall \, j=0,1,\ldots , d-1\) and it does not contain the point \(P^{\nu _{d,\varvec{\sigma }}}\). \(\square \)
Corollary 2.7
Any \(d+1\) points of \(\mathcal {V}_{d,\varvec{\sigma }},\) \(d \ge 2,\) are in general position.
Proof
It is enough to take the \(\Pi _i\)’s of dimension 0. \(\square \)
Corollary 2.8
A set of \(d+2\) linearly dependent points of \(\mathcal {V}_{d,\varvec{\sigma }}\) is the \((d,\varvec{\sigma })\)-Veronese embedding of points contained in a line of \({{\,\mathrm{{PG}}\,}}(n-1,q^t)\).
Proof
The statement needs to be proved for \(n>2\). Let \(P_0,P_1,\ldots ,P_d,P_{d+1}\) be \(d+2\) points whose embedding is linearly dependent. Let \(\Pi _i:=P_i\) for \(i=2,\ldots ,d+1\) and let \(\Pi _1=\langle P_0,P_{1} \rangle \). Suppose that \(P_i\notin \Pi _1\), with \(i=2,\ldots ,d+1\), then by Theorem 2.6,but by hypothesisa contradiction. \(\square \)
$$\begin{aligned} P_i^{\nu _{d,\varvec{\sigma }}} \notin \left\langle \Pi _1^{\nu _{d,\varvec{\sigma }}},\Pi _2^{\nu _{d,\varvec{\sigma }}},\ldots ,\Pi _{i-1}^{\nu _{d,\varvec{\sigma }}}, \Pi _{i+1}^{\nu _{d,\varvec{\sigma }}},\ldots ,\Pi _{d+1}^{\nu _{d,\varvec{\sigma }}} \right\rangle , \end{aligned}$$
$$\begin{aligned} P_i^{\nu _{d,\varvec{\sigma }}}{} & {} \in \left\langle P_0^{\nu _{d,\varvec{\sigma }}},P_1^{\nu _{d,\varvec{\sigma }}},\ldots ,P_{i-1}^{\nu _{d,\varvec{\sigma }}},P_{i+1}^{\nu _{d,\varvec{\sigma }}},\ldots , P_{d+1}^{\nu _{d,\varvec{\sigma }}}\right\rangle \subset \left\langle \Pi _1^{\nu _{d,\varvec{\sigma }}},\Pi _2^{\nu _{d,\varvec{\sigma }}}, \ldots ,\Pi _{i-1}^{\nu _{d,\varvec{\sigma }}},\Pi _{i+1}^{\nu _{d,\varvec{\sigma }}},\right. \\{} & {} \quad \left. \ldots ,\Pi _{d+1}^{\nu _{d,\varvec{\sigma }}}\right\rangle , \end{aligned}$$
In order to prove the next Corollary, we need the following
Lemma 2.9
[8] Let \(d<|{\mathbb {K}}|\). Let S be a set of \(d+2\) subspaces of \({{\,\mathrm{{PG}}\,}}(2d-1,{\mathbb {K}})\) of dimension \(d-1,\) pairwise disjoint, linearly dependent as points of the Grassmannian and such that any \(d+1\) elements of S are linearly independent. Then a line intersecting 3 elements of S intersects all of them.
Since we have assumed \(|\varvec{\sigma }|<q^t\), Lemma 2.9 always applies to \(\mathcal {V}_{d,\varvec{\sigma }}\).
Corollary 2.10
A set of \(d+2\) linearly dependent points of \(\mathcal {V}_{d,\varvec{\sigma }}\) is the Grassmann embedding of \((d-1)\)-subspaces of the normal rational scroll \(S_{1,1,\ldots ,1}\subset {{\,\mathrm{{PG}}\,}}(2d-1,q^t)\) such that a line intersecting 3 of them must intersect all of them.
Proof
By Corollary 2.8, a set \(\{P_0^{\nu _{d,\varvec{\sigma }}},P_1^{\nu _{d,\varvec{\sigma }}},\ldots ,P_{d+1}^{\nu _{d,\varvec{\sigma }}}\}\) of \(d+2\) linearly dependent points of \(\mathcal {V}_{d,\varvec{\sigma }}\) is such that \(P_0,P_1,\ldots ,P_{d+1}\) are contained in the same line, hence \(\{P_0^{\nu _{d,\varvec{\sigma }}},P_1^{d,\nu _{\varvec{\sigma }}},\ldots ,P_{d+1}^{\nu _{d,\varvec{\sigma }}}\}\) is contained in a variety \(\mathcal {V}_{d,\varvec{\sigma }}\) of dimension 1.
Hence, \(\{P_0^{\nu _{d,\varvec{\sigma }}},P_1^{\nu _{d,\varvec{\sigma }}},\ldots ,P_{d+1}^{\nu _{d,\varvec{\sigma }}}\}\) is the Grassmann embedding of the \((d-1)\)-subspaces of the normal rational scroll \(S_{1,1,\ldots ,1}\subset {{\,\mathrm{{PG}}\,}}(2d-1,q^t)\). Then the result follows from Corollary 2.7 and Lemma 2.9. \(\square \)
Theorem 2.11
A set of \(d+2\) linearly dependent points of \(\mathcal {V}_{d,\varvec{\sigma }}\) is the \(\varvec{\sigma }\)-Veronese embedding of points on a subline \(\cong {{\,\mathrm{{PG}}\,}}(1,q'),\) where \({\mathbb {F}}_{q'}\) is the largest subfield of \({\mathbb {F}}_{q^t}\) fixed by \(\sigma _i\) in \(\varvec{\sigma }\).
Proof
Let \( \langle u_i\otimes u_i^{\sigma _1}\otimes \cdots \otimes u_i^{\sigma _{d-1}} \rangle \), \(i=0,1,\ldots ,d+1\) be \(d+2\) linearly dependent points of \(\mathcal {V}_{d,\varvec{\sigma }}\), and by Corollary 2.8, we can assume \(\mathcal {V}_{d,\varvec{\sigma }}\) to be of dimension 1. Then, embed \({{\,\mathrm{{PG}}\,}}(1,q^t)\) as the subspace of \({{\,\mathrm{{PG}}\,}}(2d-1,q^t)\) spanned by \( \langle e_0 \rangle , \langle e_1 \rangle \), say \(\Pi \), and hence we can writeWe stress out that \(\phi ^j\) and \(\sigma _j\) commute and that the vectors \(u_i\)’s are pairwise not proportional. Let \(S_i:=\langle u_i,u_i^{\phi \sigma _1},\ldots ,u_i^{\phi ^{d-1} \sigma _{d-1}} \rangle \), for all \(i=0,1,\ldots ,d+1\), so we observe that \(S_i\cap S_j= \emptyset \,\, \forall \,\, i \ne j\). Then take a point \(P \in S_0\) such that \(P \notin \langle \Pi ^{\phi ^h}, h \ne j \rangle \) for any fixed \(j \in \{0,1,\ldots ,d-1\}\). The subspace \(\langle P,S_1 \rangle \) intersects \(S_2\) in a point, say R. Let \(\ell \) be the line spanned by P and R. Then \(\ell \) has non empty intersection with \(S_1\) as well. Hence, by Corollary 2.10, \(\ell \) has non empty intersection with all the \(S_i\)’s. By the choice of P, the line \(\ell \) is not contained in any \(\langle \Pi ^{\phi ^h}, h \ne j \rangle \) for a fixed \(j \in \{0,1,\ldots ,d-1\}\). If \(\ell \) intersects \(\langle \Pi ^{\phi ^h}, h \ne j \rangle \) for some \(j \in \{0,1,\ldots ,d-1\}\), then it would be projected to a unique point of \(\Pi ^{\phi ^j}\) from \(\langle \Pi ^{\phi ^h}, h \ne j \rangle \). Since \(u_i\ne u_h\) \(\forall i\ne h\), then \(u_i^{\sigma _j}\ne u_h^{\sigma _j}\) \(\forall i\ne h\) and \(\ell \) can be projected on a unique point only if \(\ell \cap S_i\) is in \(\langle \Pi ^{\phi ^h}, h \ne j \rangle \) for all the \(S_i's\) except one, a contradiction. Indeed, the point \(\ell \cap S_i= \langle \lambda _0u_i+\lambda _1u_i^{\phi \sigma _1}+\ldots +\lambda _{d-1}u_i^{\phi ^{d-1}\sigma _{d-1}} \rangle \) and the projection of \(\ell \cap S_i\) over \(\Pi ^{\phi ^j}\) is the point \(\langle u_i^{\phi ^j\sigma _j} \rangle \), so \(h_j\) cannot be zero. Therefore, \(\ell \cap \langle \Pi ^{\phi ^h}, h \ne j \rangle =\emptyset \) for any fixed \(j \in \{0,1,\ldots ,r-1\}\). Hence the projection of \(\ell \) on a \(\Pi ^{\phi ^j}\) is an isomorphism of lines, say \(p_j\) and \((\ell \cap S_i)^{p_j}= \langle u_i^{\phi ^j\sigma _j}\rangle \). By \((\ell \cap S_i)^{p_j\phi ^{-j}}=(\ell \cap S_i)^{p_0\sigma _j}\) we get that \((\ell \cap S_i)^{p_0}\) is fixed by the semi-linear collineation \(\sigma _j\phi ^jp^{-1}_jp_0\). If a semi-linear collineation of \(\Pi \cong {{\,\mathrm{{PG}}\,}}(1,q^t)\) fixes at least 3 points, then it fixes a subline \(\cong {{\,\mathrm{{PG}}\,}}(1,q')\), where \({\mathbb {F}}_{q'}\) is the subfield of \({\mathbb {F}}_{q^t}\) fixed by \(\sigma _j\). This is true for all \(\sigma _j\) in \(\varvec{\sigma }\). \(\square \)
$$\begin{aligned} u_i\otimes u_i^{\sigma _1}\otimes \cdots \otimes u_i^{\sigma _{d-1}}=u_i\wedge u_i^{\phi \sigma _1}\wedge \cdots \wedge u_i^{\phi ^{d-1}\sigma _{d-1}}. \end{aligned}$$
Finally, since the algebraic variety \(\Sigma _{(n-1)^d}\) has dimension \(d(n-1)\) and degree \(\left( {\begin{array}{c}d(n-1)\\ n-1,n-1,\cdots ,n-1\end{array}}\right) =\frac{(d(n-1))!}{d(n-1)!}\), a general subspace of \({{\,\mathrm{{PG}}\,}}(N-1,q^t)\) of codimension \(d(n-1)\) contains at most \(\frac{(d(n-1))!}{d(n-1)!}\) points of \(\mathcal {V}_{d,\varvec{\sigma }}\).
Moreover, the Segre variety is smooth and hence the tangent space \(T_P(\Sigma _{(n-1)^d})\) to \(\Sigma _{(n-1)^d}\) at a point \(P= \langle v_0\otimes v_1\otimes \cdots \otimes v_{d-1} \rangle \) has dimension \(d(n-1)\) and it spanned by the d subspacesThese subspaces pairwise intersect only in P and they are the maximal subspaces contained in \(\Sigma _{(n-1)^d}\) through the point P, and \(\Sigma _{(n-1)^d}\) does not share with \(\mathcal V_{d,\varvec{\sigma }}\) the property proved in Corollary 2.7. We have, in fact, \(T_P(\Sigma _{(n-1)^d})\cap \mathcal V_{d,\varvec{\sigma }}=P\) for each \(P \in \mathcal V_{d,\varvec{\sigma }}\).
$$\begin{aligned} \langle \langle v_0 \otimes v_1 \otimes \cdots v_{i-1} \otimes u_{i} \otimes v_{i+1} \otimes \cdots \otimes v_{d-1} \rangle \,|\, \langle u_i\rangle \in {{\,\mathrm{{PG}}\,}}(n-1,q^t)\rangle \cong {{\,\mathrm{{PG}}\,}}(n-1,q^t). \end{aligned}$$
3 The code \(\mathcal {C}_{d,\varvec{\sigma }}\)
As we have seen in Example 2.3, the SLP-variety turns out to be a variety of a subgeometry of order q, even though the array \(\varvec{\sigma }\) is defined on a finite field of order \(q^t\), hence among all the possible choice of \(\varvec{\sigma }\) and n, for q ‘big enough’ \(\mathcal V_{t,\sigma }\) is the variety with the most ‘dense’ set of points of a projective space with the property that any \(d+1\) points are independent. In this case, since \(d=t\) and, as proved in [3], \(t+2\) linearly dependent points are contained in a normal rational curve of degree t of \({{\,\mathrm{{PG}}\,}}(t,q)\), \(q>t\).
For the classical Veronese variety of degree d, hence for \(\varvec{\sigma }={\textbf {1}}\), Theorem 2.11 implies that \(d+2\) linearly dependent points are contained in the Veronese embedding of degree d of a line, hence in a normal rational curve of degree d of \({{\,\mathrm{{PG}}\,}}(d,q^t)\).
Finally, for a general \((d,\varvec{\sigma })\)-Veronese variety, if \(d+2>q'+1\), with \(q'\) defined as in Corollary 2.11, every \(d+2\) points of \(\mathcal {V}_{d,\varvec{\sigma }}\) are linearly independent, hence, for ‘small’ \(q'\), it provides a dense set of points with that property. More precisely, we get \(\frac{q^{nt}-1}{q^t-1}\) points in \({{\,\mathrm{{PG}}\,}}(N-1,q^t)\) such that any \(d+2\) of them are in general position. Sets of points with properties of this sort are studied for their connections with linear codes.
If H is the matrix whose columns are the coordinates vectors of the points of the variety \(\mathcal V_{d,\varvec{\sigma }}\), we get a code \({\mathcal C}_{d,\varvec{\sigma }}\) and we may study the minimum distance of it and characterize the codewords of minimum weight (for an overview on this topic, see, e.g., [1]).
Definition 3.1
Let \(\mathcal V_{d,\varvec{\sigma }}\) be a \((d,\varvec{\sigma })\)-Veronese variety and denote by \(\mathcal {C}_{d,\varvec{\sigma }}\) the code whose parity check matrix H of order \(N\times \left( \frac{q^{nt}-1}{q^t-1}\right) \) has columns that are the coordinate vectors of the points of the variety \(\mathcal V_{d,\varvec{\sigma }}\).
Clearly, the order of the columns of H is arbitrary, so that Definition 3.1 makes sense only up to code equivalence, as a permutation of the columns that is not usually an automorphism of the code, see [3, Remark 3.3].
Definition 3.2
The support of a codeword \( {\textbf {w}} \in {\mathcal C}_{d,\varvec{\sigma }}\) is the set of the points of the variety \(\mathcal V_{d,\varvec{\sigma }}\) corresponding to the non-zero positions of \({\textbf {w}}\).
As showed in [3, Theorem 3.5], the following result holds
Theorem 3.3
Let \(\varvec{\sigma }\in G^d\) with \(d_{\varvec{\sigma }}=(d_0,d_1,\ldots ,d_m),\) \(|\varvec{\sigma }|<q^t\) and \({\mathbb F}_{q'}\) be the largest subfield fixed by \(\sigma _i\)’s. If \(d<q'\) then the code \({\mathcal C}_{d,\varvec{\sigma }}\) has length \(r=\frac{q^{nt}-1}{q^t-1}\) and parameters \([r,r-N,d+2]\).
Proof
Since \(|\mathcal V_{d,\varvec{\sigma }}|=|{{\,\mathrm{{PG}}\,}}(n - 1, q^t )|\) the code \({\mathcal C}_{d,\varvec{\sigma }}\) has lenght \(\frac{q^{nt}-1}{q^t-1}\) . Moreover, since \(\mathcal V_{d,\varvec{\sigma }}\) is not contained in any hyperplane of \({{\,\mathrm{{PG}}\,}}(N-1,q^t)\), the vector space dimension of the \(N \times r\) matrix H is maximal and so the dimension of the code is \(r - N\). By Corollary 2.7 guarantees that any \( d + 1\) columns of H are linearly independent; thus, by [10, Theorem 10, p. 33], the minimum distance of \({\mathcal C}_{d,\varvec{\sigma }}\) is at least \(d + 2\). The image under \(\nu _{d,\varvec{\sigma }}\) of the canonical subline \({{\,\mathrm{{PG}}\,}}(1, q')\) of \({{\,\mathrm{{PG}}\,}}(n - 1, q^t)\) determines a submatrix \(H'\) of H with many repeated rows; indeed, the points represented in H constitute a normal rational curve \({{\,\mathrm{{PG}}\,}}(d,q')\) and it follows that any \(d + 2\) such points are necessarily dependent. Hence, the minimum distance is exactly \(d+ 2\). \(\square \)
Now, as in [3, Theorem 3.7], by the characterizations of sets of \(d+2\) points of \(\mathcal V_{d,\varvec{\sigma }}\) which are linearly dependent yields a characterization of the minimum weight codewords of the associated code. More precisely,
Theorem 3.4
A codeword \(\text {{\textbf {w}}} \in {\mathcal C}_{d,\varvec{\sigma }}\) has minimum weight if and only if its support consists of \(d+2\) points contained in the image of a subline \({{\,\mathrm{{PG}}\,}}(1,q'),\) \(d<q',\) where \({\mathbb F}_{q'}\) is the largest subfield of \({\mathbb F}_{q^t}\) fixed by \(\sigma _i\) for all \(\sigma _i\) in \(\varvec{\sigma }\).
Suppose \(d \ge q'\) where \({\mathbb {F}}_{q'}\) is the largest subfield of \({\mathbb {F}}_{q^t}\) fixed by \(\sigma _i\), for all \(\sigma _i\) in \(\varvec{\sigma }\). By Theorem 2.11, the code \({\mathcal C}_{d,\varvec{\sigma }}\) is a linear code with minimum distance \( d+3 \le \delta \le N+1\). If the Singleton bound is reached, then it is an MDS code. Let N be as in (4) with \(\displaystyle \sum \nolimits _{i=0}^{m}d_i=d\). If \(n=2\), thenand the minimum is reached for \(m=1,d_0=d-1,d_1=1\), so \(N=2d\). If \(\sigma \) is such that \(\textrm{Fix}(\sigma )\cap {\mathbb {F}}_{q^t}={\mathbb {F}}_p\), where p is the characteristic of the field, since we should have \(d\ge p\), the smallest possible \(d=p\) and in this casegetting that \(\mathcal {V}_{d,\varvec{\sigma }}\) is a set of \(q^t+1\) points in \({{\,\mathrm{{PG}}\,}}(2p-1,q^t)\) such that any \(p+2\) of them are in general position. So the code \({\mathcal C}_{d,\varvec{\sigma }}\) is a \([q^t+1,q^t-2p+1]\)-linear code with minimum distance at least \(p+3\) and the Singleton bound \(2p+1\). Now, if \(\sigma : x \mapsto x^p\), then \(\mathcal V_{p,\varvec{\sigma }}\) is the normal rational curve of \({{\,\mathrm{{PG}}\,}}(2p-1,q^t)\); hence \({\mathcal C}_{p,\varvec{\sigma }}\) is an MDS code.
$$\begin{aligned} N=\displaystyle \prod _{i=0}^{m}(d_i+1) \end{aligned}$$
$$\begin{aligned} \varvec{\sigma }=(\underbrace{1,1,\ldots ,1}_{p-1\, \text {times}},\sigma ) \end{aligned}$$
(5)
Furthermore for \(p\in \{2,3\}\), the following cases can also occurClearly, as p gets larger, the minimum distance gets smaller than the Singleton bound. Before showing the announced result, we recall the following theorem due to Thas [15] and of which Kaneta and Maruta gives an elementary proof,
-
for \(p=2\), \(\sigma : x \mapsto x^{2^h}\), \(1< h < et\), \(\mathcal V_{2,\varvec{\sigma }}\) is either the Segre arc or the normal rational curve (for \(h=et-1\)), hence \({\mathcal C}_{2,\varvec{\sigma }}\) is an MDS code,
Theorem 3.5
[7, Theorem 1] In \({{\,\mathrm{{PG}}\,}}(r,q),\) \(r\ge 2\) and q odd, every k-arc withis contained in one and only one normal rational curve of the space. In particular, if \(q > (4r-5)^2,\) then every \((q+1)\)-arc is a normal rational curve.
$$\begin{aligned} q -\sqrt{q}/4 + r-1/4 \le k \le q+1 \end{aligned}$$
Theorem 3.6
Let \(q=3^e\) and \(\sigma : x \in {\mathbb F}_{q^t} \mapsto x^{3^h} \in {\mathbb F}_{q^t},\) \(1<h<et,\) \(\gcd (h,et)=1\) with \(et>4\). Then \(\mathcal V_{3,\varvec{\sigma }}\) with \(\varvec{\sigma }=(1,1,\sigma )\) is a \((3^{et}+1)\)-track of \({{\,\mathrm{{PG}}\,}}(5,3^{et})\) and \({\mathcal C}_{3,\varvec{\sigma }}\) is an almost MDS.
Proof
By the previous considerations, since the \([q^t+1,q^t-5]\)-code \({\mathcal C}_{d,\varvec{\sigma }}\) has distance at least 6, the result follows showing the existence of 6 columns of H linearly dependent or equivalently that there exists 6 points linearly dependent of the setSuppose that any 6 points of \(\mathcal V_{3,\varvec{\sigma }}\) with \(\varvec{\sigma }=(1,1,\sigma )\) are linearly independent, hence \(\mathcal V_{3,\varvec{\sigma }}\) is an arc of \({{\,\mathrm{{PG}}\,}}(5,q^t)\). By Theorem 3.5, \(\mathcal V_{3,\varvec{\sigma }}\) must be projectively equivalent to rational normal curveSince the normal rational curve has a 3-transitive automorphisms group, we can always assume that there is a collineation of \({{\,\mathrm{{PG}}\,}}(5,q^t)\) fixing (0, 0, 0, 0, 0, 1) and (1, 0, 0, 0, 0, 0). Moreover, w.l.o.g. we can assume that this collineation has the identity as companion automorphism.
$$\begin{aligned} \mathcal V_{3,\varvec{\sigma }}=\left\{ \left( 1,z,z^2,z^{3^h},z^{3^h+1},z^{3^h+2}\right) \,:\, z \in {\mathbb F}_{q^t}\right\} \cup \{(0,0,0,0,0,1)\}. \end{aligned}$$
$$\begin{aligned} \left\{ (1,y,y^2,y^{3},y^{4},y^{5}) \,:\, y \in {\mathbb F}_{q^t}\right\} \cup \{(0,0,0,0,0,1)\}. \end{aligned}$$
Hence there must be \(f_i(y)\in {\mathbb F}_{q^t}[y]\) of degree at most 5 and linearly independent such thatwith \(f_i(y)\) vanishing in 0 for \(i \in \{1,2,3,4,5\}\) and \(f_0(0)=1\) up to a nonzero scalar. So, \(f_0(y)=1\) for all \(y \in {\mathbb F}_{q^t}\) and since \(\deg f_0(y)\le 5 < q^t\), then \(f_0(y)=1\). Note that \(\deg f_i(y) \ne 0\) for \(i=1,2,3,4,5\) andbut \(2\deg f_1(y)\le 10 < q^t\), and hence \(f_2(y)=f_1(y)^2\) and \(\deg f_1(y)\le 2\). Similarly,but \(3^h\deg f_1(y)\le 3^h\cdot 2 < q^t\), so \(f_4(y)=f_1(y)^{3^h}\) and \(3^h \deg f_1(y) \le 5\), obtaining \(3^h \le 5\), a contradiction. \(\square \)
$$\begin{aligned} (f_0(y),f_1(y), f_2(y),f_3(y),f_4(y),f_5(y))=\left( 1,z,z^2,z^{3^h},z^{3^h+1},z^{3^h+2}\right) \end{aligned}$$
$$\begin{aligned} f_2(y)=f_1(y)^2 \quad \mod \,\, y^{q^t}-y, \end{aligned}$$
$$\begin{aligned} f_4(y)=f_1(y)^{3^h}\quad \mod \,\,y^{q^t}-y, \end{aligned}$$
Actually, the result above holds for \(q^{t}=27,81\) as well, this is verified by the software MAGMA, obtaining an infinite family of almost MDS codes or, equivalently, an infinite family of \((3^{et}+1)\)-tracks of \({{\,\mathrm{{PG}}\,}}(5,3^{et})\) with \(et>2\).
Acknowledgements
The authors thank Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM) for having supported this research.
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