Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries

TL;DR

This paper constructs infinite families of non-linear maximum rank distance codes using bilinear forms and finite geometry models, extending recent results and providing a geometric perspective on these codes.

Contribution

It introduces new infinite families of non-linear maximum rank distance codes via geometric methods and relates them to existing codes in the literature.

Findings

Constructed infinite families of non-linear maximum rank distance codes.

Provided a geometric description using the cyclic model for field reduction.

Unified and extended previous non-linear code constructions.

Abstract

In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino and Pavese.

Equations

$$|\mathcal{X}|\le q^{m^{\prime }(m-s)}.$$

$$\mathrm{rk}(f)=m-{\dim }_{{\mathbb{F}}_q}(\mathrm{Rad}(f)).$$

$$\begin{array}[]{rccc}\nu=\nu_{\{u_{1},\ldots,u_{m}\}}:&\Omega&\rightarrow&M_{m,m}(\mathbb{F}_{q})\\ &f&\mapsto&(f(u_{i},u_{j}))_{i,j=1,\ldots,m}\end{array}$$

$$f^{(g,g^{\prime })}(v,v^{\prime })=f(gv,g^{\prime }{v}^{\prime }).$$

$$f^{\top }(v,v^{\prime })=f(v^{\prime },v),$$

$$V=\left\{\sum _{j=1}^m{a}^{q^{j-1}}{v}_j:a\in {\mathbb{F}}_{q^m}\right\}.$$

$$V^{\ast }=\left\{\sum _{i=1}^m{\alpha }^{q^{i-1}}{v}_i^{\ast }:\alpha \in {\mathbb{F}}_{q^m}\right\}.$$

$$D_{(a_0,a_1,\dots ,a_{m-1})}=\left(\begin{array}{cccc}{a}_0& a_1& \cdots & a_{m-1}\\ a_{m-1}^q& a_0^q& \cdots & a_{m-2}^q\\ ⋮& ⋮& \ddots & ⋮\\ a_1^{q^{m-1}}& a_2^{q^{m-1}}& \cdots & a_0^{q^{m-1}}\end{array}\right)$$

$$L(x)=\sum _{i=0}^{m-1}{\alpha }_i{x}^{q^i},{\alpha }_i\in {\mathbb{F}}_{q^m},$$

$$\begin{array}[]{cccc}\varphi:&\mathcal{L}_{m}(\mathbb{F}_{q^{m}})&\longrightarrow&\mathcal{D}_{m}(\mathbb{F}_{q^{m}})\\ &\sum_{i=0}^{m-1}{\alpha_{i}x^{q^{i}}}&\longmapsto&D_{(\alpha_{0},\ldots,\alpha_{m-1})}\end{array}$$

$$f_{\mathbf{a}}(v_i,v_j)=a_{m-i+j}^{q^{i-1}}\mathrm{for}i,j=1,\dots ,m$$

$$\mathrm{Tr}:y\in {\mathbb{F}}_{q^m}\to \mathrm{Tr}(y)=\sum _{j=0}^{m-1}{y}^{q^j}\in {\mathbb{F}}_q.$$

$$f_{\mathbf{a}}(v,v^{\prime })=f_{\mathbf{a}}(x,x^{\prime })=\mathrm{Tr}(L_{\mathbf{a}}(x^{\prime })x),$$

$${\mathcal{F}}_{\mathcal{A}}=\{f_{\mathbf{a}}\in \Omega :\mathbf{a}\in \mathcal{A}\}.$$

$$\begin{array}[]{rccc}\nu_{\{v_{1},\ldots,v_{m}\}}:&\Omega&\rightarrow&\mathcal{D}_{m}(\mathbb{F}_{q^{m}})\\ &f_{\mathbf{a}}&\mapsto&D_{(a_{0},\ldots,a_{m-1})}\end{array}$$

$$N:x\in {\mathbb{F}}_{q^m}\to N(x)=\prod _{j=0}^{m-1}{x}^{q^j}\in {\mathbb{F}}_q.$$

$${\pi }_{\alpha }=\{(\lambda x,\lambda \alpha x^q,\lambda {\alpha }^{1+q}{x}^{q^2},\dots ,\lambda {\alpha }^{1+q+\dots +q^{m-2}}{x}^{q^{m-1}}):\lambda ,x\in {\mathbb{F}}_{q^m}\setminus \{0\}\}.$$

$${\alpha }^{1+q+\dots +q^k}=c^{-1}{\beta }^{1+q+\dots +q^k}{c}^{q^{k+1}}.$$

$$\begin{array}[]{rcl}1&=&\lambda x\\ \alpha&=&\lambda\beta x^{q}\\ \alpha^{1+\ldots+q^{m-2}}&=&\lambda\beta^{1+\ldots+q^{m-2}}x^{q^{m-1}}\end{array}$$

$${\alpha }^{q+q^2+\dots +q^{m-1}}={\lambda }^q{\beta }^{q+q^2+\dots +q^{m-1}}x.$$

$$N(\alpha )={\lambda }^q\lambda N(\beta )x{x}^q=N(\beta ).$$

$$(\lambda x,\lambda \alpha x^q,\lambda {\alpha }^{1+q}{x}^{q^2},\dots ,\lambda {\alpha }^{1+\dots +q^{m-2}}{x}^{q^{m-1}})=(\rho y,\rho \alpha y^q,\rho {\alpha }^{1+q}{y}^{q^2},\dots ,\rho {\alpha }^{1+\dots +q^{m-2}}{x}^{q^{m-1}})$$

$$\begin{array}[]{lcl}(1-\lambda x)z_{i}+(\alpha-\lambda\beta x^{q})z_{i}^{q}+\ldots+(\alpha^{1+\ldots+q^{m-2}}-\lambda\beta^{1+\ldots+q^{m-2}}x^{q^{m-1}})z_{i}^{q^{m-1}}&=&0\end{array}$$

$$\begin{array}[]{rcl}(\alpha^{q+\ldots+q^{m-1}}-\lambda^{q}\beta^{q+\ldots+q^{m-1}}x)z_{i}+(1-\lambda^{q}x^{q})z_{i}^{q}+\\[3.61371pt] \ldots+(\alpha^{q+\ldots+q^{m-2}}-\lambda^{q}\beta^{q+\ldots+q^{m-2}}x^{q^{m-1}})z_{i}^{q^{m-1}}&=&0,\end{array}$$

$$\begin{array}[]{rcl}[a-1+(\lambda-\lambda^{q}\alpha\beta^{q+\ldots+q^{m-1}})x]z_{i}+(\lambda\beta-\lambda^{q}\alpha)x^{q}z_{i}^{q}+\\[3.61371pt] \ldots+(\lambda\beta-\lambda^{q}\alpha)\beta^{q+\ldots+q^{m-2}}x^{q^{m-1}}z_{i}^{q^{m-1}}&=&0,\end{array}$$

$$(a-1+(\lambda -{\lambda }^q\alpha {\beta }^{q+\dots +q^{m-1}})x,(\lambda \beta -{\lambda }^q\alpha )x^q,\dots ,(\lambda \beta -{\lambda }^q\alpha ){\beta }^{q+\dots +q^{m-2}}{x}^{q^{m-1}})$$

$$\left\{\begin{array}[]{lcl}z_{1}X_{1}+z_{1}^{q}X_{2}+\ldots+z_{1}^{q^{m-1}}X_{m}&=&0\\ z_{2}X_{1}+z_{2}^{q}X_{2}+\ldots+z_{2}^{q^{m-1}}X_{m}&=&0\end{array}\right.$$

$$x_1=-{\Delta }^{-1}\left(\left|\begin{array}{cc}{z}_1^{q^2}& z_1^q\\ z_2^{q^2}& z_2^q\end{array}\right|x_3+\left|\begin{array}{cc}{z}_1^{q^3}& z_1^q\\ z_2^{q^3}& z_2^q\end{array}\right|x_4+\dots +\left|\begin{array}{cc}{z}_1^{q^{m-1}}& z_1^q\\ z_2^{q^{m-1}}& z_2^q\end{array}\right|x_m\right)$$

$$x_2=-{\Delta }^{-1}\left(\left|\begin{array}{cc}{z}_1& z_1^{q^2}\\ z_2& z_2^{q^2}\end{array}\right|x_3+\left|\begin{array}{cc}{z}_1& z_1^{q^3}\\ z_2& z_2^{q^3}\end{array}\right|x_4+\dots +\left|\begin{array}{cc}{z}_1& z_1^{q^{m-1}}\\ z_2& z_2^{q^{m-1}}\end{array}\right|x_m\right).$$

$$c_i=\left|\begin{array}{cc}{z}_1& z_1^{q^{i-1}}\\ z_2& z_2^{q^{i-1}}\end{array}\right|,i=3,\dots ,m;$$

Full text

1 Introduction

Let $M_{m,m^{\prime}}(\mathbb{F}_{q})$, $m\leq m^{\prime}$, be the rank metric space of all the $m\times m^{\prime}$ matrices with entries in the finite field $\mathbb{F}_{q}$ with $q$ elements, $q=p^{h}$, $p$ a prime. The distance between two matrices by definition is the rank of their difference. An $(m,m^{\prime},q;s)$-rank distance code (also rank metric code) is any subset $\mathcal{X}$ of $M_{m,m^{\prime}}(\mathbb{F}_{q})$ such that the minimum distance between two of its distinct elements is $s+1$. An $(m,m^{\prime},q;s)$-rank distance code is said to be linear if it is a linear subspace of $M_{m,m^{\prime}}(\mathbb{F}_{q})$.

It is known [11] that the size of an $(m,m^{\prime},q;s)$-rank distance code $\mathcal{X}$ is bounded by the Singleton-like bound:

[TABLE]

When this bound is achieved, $\mathcal{X}$ is called an $(m,m^{\prime},q;s)$-maximum rank distance code, or $(m,m^{\prime},q;s)$-MRD code, for short.

Although MRD codes are very interesting by their own and they caught the attention of many researchers in recent years [1, 9, 32], such codes have also applications in error-correction for random network coding [18, 22, 37], space-time coding [38] and cryptography [17, 36].

Obviously, investigations of MRD codes can be carried out in any rank metric space isomorphic to $M_{m,m^{\prime}}(\mathbb{F}_{q})$. In his pioneering paper [11], Ph. Delsarte constructed linear MRD codes for all the possible values of the parameters $m$, $m^{\prime}$, $q$ and $s$ by using the framework of bilinear forms on two finite-dimensional vector spaces over a finite field (Delsarte used the terminology Singleton systems instead of maximum rank distance codes).

Few years later, Gabidulin [16] independently constructed Delsarte’s linear MRD codes as evaluation codes of linearized polynomials over a finite field [26]. That construction was generalized in [21] and these codes are now known as Generalized Gabidulin codes.

In the case $m^{\prime}=m$, a different construction of Delsarte’s MRD codes was given by Cooperstein [7] in the framework of the tensor product of a vector space over $\mathbb{F}_{q}$ by itself. Very recently, Sheekey [35] and Lunardon, Trombetti and Zhou [28] provide some new linear MRD codes by using linearized polynomials over $\mathbb{F}_{q^{m}}$.

In finite geometry, $(m,m,q;m-1)$-MRD codes are known as spread sets [12]. To the extent of our knowledge the only non-linear MRD codes that are not spread sets are the $(3,3,q;1)$-MRD codes constructed by Cossidente, Marino and Pavese in [8]. They got such codes by looking at the geometry of certain algebraic curves of the projective plane ${\rm PG}(2,q^{3})$. Such curves, called $C_{F}^{1}$-sets, were introduced and studied by Donati and Durante in [13]. In this paper, we construct infinite families of non-linear $(m,m,q;m-2)$-MRD codes, for $q\geq 3$ and $m\geq 3$. We also show that the Cossidente, Marino and Pavese non-linear MRD codes belong to these families. Our investigation will carry out in the framework of bilinear forms on a finite dimensional vector space over $\mathbb{F}_{q}$.

Let $\Omega=\Omega(V,V)$ be the set of all bilinear forms on $V$, where $V=V(m,q)$ denotes an $m$-dimensional vector space over $\mathbb{F}_{q}$. Clearly, $\Omega$ is an $m^{2}$-dimensional vector space over $\mathbb{F}_{q}$.

The *left radical * ${\rm Rad}(f)$ of any $f\in\Omega$ by definition is the subspace of $V$ consisting of all vectors $v$ satisfying $f(v,v^{\prime})=0$ for every $v^{\prime}\in V$. The rank of $f$ is the codimension of ${\rm Rad}(f)$, i.e.

[TABLE]

Let $u_{1},\ldots,u_{m}$ be a basis of $V$. For a given $f\in\Omega$, the matrix $(f(u_{i},u_{j}))_{i,j=1,\ldots,m}$, is called the *matrix * of $f$ in the basis $u_{1},\ldots,u_{m}$ and the map

[TABLE]

is an isomorphism of rank metric spaces giving ${\rm rk}(f)={\rm rk}(\nu(f))$.

The group $H={\rm GL}(V)\times{\rm GL}(V)$ acts on $\Omega$ as a subgroup of ${\rm Aut}_{\mathbb{F}_{q}}(\Omega)$: for every $(g,g^{\prime})\in H$, the $(g,g^{\prime})-$image of any $f\in\Omega$ is defined to be the bilinear form $f^{(g,g^{\prime})}$ given by

[TABLE]

Any $\theta\in{\rm Aut}(\mathbb{F}_{q})$ naturally defines a semilinear transformation of $V$. For any $f\in\Omega$ and $\theta\in{\rm Aut}(\mathbb{F}_{q})$, we can define the bilinear form $f^{\theta}(v,v^{\prime})=f(v^{\theta^{-1}},{v^{\prime}}^{\theta^{-1}})^{\theta}$.

The involutorial operator $\top:f\in\Omega\rightarrow f^{\top}\in\Omega$, where $f^{\top}$ is given by

[TABLE]

is an automorphism of $\Omega$. It turns out that the above automorphisms are all the elements in ${\rm Aut}_{\mathbb{F}_{q}}(\Omega)$, i.e. ${\rm Aut}_{\mathbb{F}_{q}}(\Omega)=({\rm GL}(V)\times{\rm GL}(V))\rtimes\langle\top\rangle\rtimes{\rm Aut}(\mathbb{F}_{q})$.

Two MRD codes $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are said to be equivalent if there exists $\varphi\in{\rm Aut}_{\mathbb{F}_{q}}(\Omega)$ such that $\mathcal{X}_{2}=\mathcal{X}_{1}^{\varphi}$.

This paper is organized as follows. In Section 2 we introduce a cyclic model of $\Omega$. In this model we construct infinite families of non-linear MRD codes. More precisely, for $q\geq 3$, $m\geq 3$ and $I$ any subset of $\mathbb{F}_{q}\setminus\{0,1\}$, we provide a subset $\mathcal{F}_{m,q;I}$ of $\Omega$ which turns out to be a non-linear $(m,m,q;m-2)$-MRD code (Theorem 2.19).

In Section 3 we give a geometric description of such codes. If a given rank distance code $\mathcal{X}$ is considered as a subset of $V(m^{2},q)$, then one can consider the corresponding set of projective points in ${\rm PG}(m^{2}-1,q)$ under the canonical homomorphism $\psi:{\rm GL}(V(m^{2},q))\rightarrow{\rm PGL}(m^{2},q)$. We prove (Theorem 3.5) that the projective set defined by $\mathcal{F}_{m,q;I}$, with $|I|=k$, is a subset of a Desarguesian $m$-spread of ${\rm PG}(m^{2}-1,q)$ [34] consisting of two spread elements, $k$ pairwise disjoint Segre varieties $\mathcal{S}_{m,m}(\mathbb{F}_{q})$ [20] and $q-1-k$ hyperreguli [30]. Additionally, if one consider the projective space ${\rm PG}(m^{2}-1,q)$ as the field reduction of ${\rm PG}(m-1,q^{m})$ over $\mathbb{F}_{q}$, then the projective set defined by $\mathcal{F}_{m,q;I}$ is, in fact, the field reduction of the union of two projective points, $k$ mutually disjoint $(m-1)$-dimensional $\mathbb{F}_{q}$-subgeometries and $q-1-k$ scattered $\mathbb{F}_{q}$-linear sets of pseudoregulus type of ${\rm PG}(m-1,q^{m})$ [13, 24, 29]. The main tool we use to get the above geometric description is the field reduction of $V(m,q^{m})$ over $\mathbb{F}_{q}$ in the cyclic model for the tensor product $\mathbb{F}_{q^{m}}\otimes V$ as described in [7].

2 The non-linear MRD codes in the cyclic model of bilinear forms

In the paper [7], the cyclic model of the $m$-dimensional vector space $V=V(m,q)$ over $\mathbb{F}_{q}$ was introduced by taking eigenvectors, say $v_{1},\ldots,v_{m}$, of a given Singer cycle $\sigma$ of $V$, where a Singer cycle of $V$ is an element of ${\rm GL}(V)$ of order $q^{m}-1$. Since the vectors $v_{1},\ldots,v_{m}$ have distinct eigenvalues over $\mathbb{F}_{q^{m}}$, they form a basis of the extension $\widehat{V}=V(m,q^{m})$ of $V$. In this basis the vector space $V$ is represented by

[TABLE]

We call $v_{1},\ldots,v_{m}$ a Singer basis of $V$ and the above representation is called the cyclic model for $V$ [19, 15].

The set of all $1-$dimensional $\mathbb{F}_{q}-$subspaces of $\widehat{V}$ spanned by vectors in the cyclic model for $V$ is called the cyclic model for the projective space ${\rm PG}(V)$. Note that the above cyclic model corresponds to the cyclic model of ${\rm PG}(V)$ where the points are identified with the elements of the group ${\mathbb{Z}}_{q^{m-1}+q^{m-2}+\cdots+q+1}$ [19, pp. 95–98] [15]. Very recently, the cyclic model for $V(3,q)$ has been used to give an alternative model for the triality quadric $Q^{+}(7,q)$ [2].

Let $\widehat{V}^{*}$ be the dual vector space of $\widehat{V}$ with basis $v^{*}_{1},\ldots,v^{*}_{m}$, the dual basis of the Singer basis $v_{1},\ldots,v_{m}$. Then the dual vector space of $V$ is

[TABLE]

A linear transformation from $V$ to itself is called an endomorphism of $V$. We will denote the set of all endomorphisms of $V$ by ${\rm End}(V)$.

An $m\times m$ Dickson matrix (or $q$-circulant matrix) over $\mathbb{F}_{q^{m}}$ is a matrix of the form

[TABLE]

with $a_{i}\in\mathbb{F}_{q^{m}}$. We say that the above matrix is generated by the array $(a_{0},a_{1},\ldots,a_{m-1})$.

Let $\mathcal{D}_{m}(\mathbb{F}_{q^{m}})$ denote the Dickson matrix algebra formed by all $m\times m$ Dickson matrices over $\mathbb{F}_{q^{m}}$. The set $\mathcal{B}_{m}(\mathbb{F}_{q^{m}})$ of all invertible Dickson $m\times m$ matrices is known as the Betti-Mathieu group [6].

Proposition 2.1**.**

[39, Lemma 4.1]** ${\rm End}(V)\simeq\mathcal{D}_{m}(\mathbb{F}_{q^{m}})$ and ${\rm GL}(V)\simeq\mathcal{B}_{m}(\mathbb{F}_{q^{m}})$.

A polynomial of the form

[TABLE]

is called a linearized polynomial (or q-polynomial) over $\mathbb{F}_{q^{m}}$. It is known that every endomorphism of $\mathbb{F}_{q^{m}}$ over $\mathbb{F}_{q}$ can be represented by a unique $q-$polynomial [33].

Let $\mathcal{L}_{m}(\mathbb{F}_{q^{m}})$ be the set of all $q$-polynomials over $\mathbb{F}_{q^{m}}$. In the paper [39], it was showed that the map

[TABLE]

is an isomorphism between the non-commutative $\mathbb{F}_{q}-$algebras $\mathcal{L}_{m}(\mathbb{F}_{q^{m}})$ and $\mathcal{D}_{m}(\mathbb{F}_{q^{m}})$. From Proposition 2.1 we see that any Singer basis of $V$ realizes this isomorphism.

Proposition 2.2**.**

Let $v_{1},\ldots,v_{n}$ be a Singer basis of $V$. Then the matrix of any $f\in\Omega$ with respect to $v_{1},\ldots,v_{n}$ is an $m\times m$ Dickson matrix. Conversely, every $m\times m$ Dickson matrix defines a bilinear form on $V\times V$.

Proof.

Let $D_{\mathbf{a}}$ be an $m\times m$ Dickson matrix generated by the $m$-ple $\mathbf{a}=(a_{0},a_{1},\ldots,a_{m-1})$ over $\mathbb{F}_{q^{m}}$. Let $f_{\mathbf{a}}$ be the bilinear mapping on $\widehat{V}\times\widehat{V}$ defined by

[TABLE]

where subscripts are taken modulo $m$, and then extended over $\widehat{V}$ by linearity. Set $L_{\mathbf{a}}(x)=\sum_{i=0}^{m-1}{a_{i}x^{q^{i}}}$ and let ${\rm Tr}$ denote the trace function from $\mathbb{F}_{q^{m}}$ onto $\mathbb{F}_{q}$:

[TABLE]

It is easily seen that the action of $f_{\mathbf{a}}$ on $V\times V$ is given by

[TABLE]

with $v=\sum_{i=1}^{m}{x^{q^{i-1}}}v_{i},v^{\prime}=\sum_{j=1}^{m}{x^{\prime q^{j-1}}}v_{j}$, which is a bilinear form on $V\times V$. The assertion follows from consideration on the size of $\mathcal{D}_{m}(\mathbb{F}_{q^{m}})$. $\Box$

For any $m$-ple $\mathbf{a}=(a_{0},\ldots,a_{m-1})$ over $\mathbb{F}_{q^{m}}$, $f_{\mathbf{a}}$ will denote the bilinear form having matrix $D_{\mathbf{a}}$ in the Singer basis $v_{1},\ldots,v_{m}$. For any set $\mathcal{A}$ of $m-$ples over $\mathbb{F}_{q^{m}}$ we put

[TABLE]

Corollary 2.3**.**

Let $\mathbf{a}=(a_{0},\ldots,a_{m-1})$. Then

[TABLE]

is an isomorphism of rank metric spaces giving ${\rm rk}(f_{\mathbf{a}})={\rm rk}(D_{(a_{0},\ldots,a_{m-1})})$.

Remark 2.4**.**

By Proposition 2.1, ${\rm Aut}_{\mathbb{F}_{q}}(\Omega)$ is represented by the group $(\mathcal{B}_{m}(\mathbb{F}_{q^{m}})\times\mathcal{B}_{m}(\mathbb{F}_{q^{m}}))\rtimes\langle t\rangle\rtimes{\rm Aut}(\mathbb{F}_{q})$ in the Singer basis $v_{1},\ldots,v_{m}$. Here, $t$ denote transposition in $M_{m,m}(\mathbb{F}_{q^{m}})$ and it corresponds to the operator $\top$.

Remark 2.5**.**

Note that $(\ref{eq_2})$ coincides with the bilinear form $(6.1)$ in [11] when $m^{\prime}=m$.

Remark 2.6**.**

Since a change of basis in $\widehat{V}\times\widehat{V}$ preserves the rank of bilinear forms, for any given $f\in\Omega$ we can consider its matrix representation in the Singer basis $v_{1},\ldots,v_{m}$. Therefore, we can assume $f=f_{\mathbf{a}}$ for some $m$-ple $\mathbf{a}$ over $\mathbb{F}_{q^{m}}$, so that ${\rm Rad}(f_{\mathbf{a}})$ is the set of vectors $v^{\prime}=x^{\prime}v_{1}+\ldots+x^{\prime q^{m-1}}v_{m}\in V$, $x^{\prime}\in\mathbb{F}_{q^{m}}$, such that $L_{\mathbf{a}}(x^{\prime})=0$.

We are now in position to construct non-linear MRD codes as subsets of $\Omega$.

Let $N$ denote the norm map from $\mathbb{F}_{q^{m}}$ onto $\mathbb{F}_{q}$:

[TABLE]

For every nonzero element $\alpha\in\mathbb{F}_{q^{m}}$, let

[TABLE]

Remark 2.7**.**

The matrix of the Singer cycle $\sigma$ of $V$ in the basis $v_{1},\ldots,v_{m}$ is ${\rm diag}(\mu,\mu^{q},\ldots,\mu^{q^{m-1}})$, where $\mu$ is a generator of the multiplicative group of $\mathbb{F}_{q^{m}}$ [7]. If $S$ is the Singer cyclic group generated by $\sigma$, then the set $\mathcal{F}_{\pi_{a}}$ is the $(S\times S)$-orbit of the bilinear form $f_{\mathbf{a}}$, with $\mathbf{a}=(1,\alpha,\alpha^{1+q},\ldots,\alpha^{1+\ldots+q^{m-2}})$. It turns out that the bilinear forms in $\mathcal{F}_{\pi_{a}}$ have constant rank.

Proposition 2.8**.**

$\pi_{\alpha}=\pi_{\beta}$* if and only if $N(\alpha)=N(\beta)$.*

Proof.

Let $\alpha,\beta\in\mathbb{F}_{q^{m}}\setminus\{0\}$ such that $N(\alpha)=N(\beta)$. By Remark 2.7 it suffices to show that $(1,\alpha,\alpha^{1+q},\ldots,\alpha^{1+q+\ldots+q^{m-2}})$ is in $\pi_{\beta}$.

Since $N(\alpha)=N(\beta)$, then $\alpha=\beta c^{q-1}$ for some $c\in\mathbb{F}_{q^{m}}\setminus\{0\}$. As

$(1+q+\ldots+q^{k})(q-1)=q^{k+1}-1$, we have

[TABLE]

Conversely, let $\pi_{\alpha}=\pi_{\beta}$. Then

[TABLE]

for some $\lambda,x\in\mathbb{F}_{q^{m}}\setminus\{0\}$. From the last equation we get

[TABLE]

By taking into account the first and second equation of (4) we get

[TABLE]

$\Box$

We will write $\pi_{a}$ instead of $\pi_{\alpha}$, if $\alpha$ is an element of $\mathbb{F}_{q^{m}}\setminus\{0\}$ with $N(\alpha)=a$.

Lemma 2.9**.**

Every $\pi_{a}$ has size $(q^{m}-1)^{2}/(q-1)$.

Proof.

Let $\alpha\in\mathbb{F}_{q^{m}}\setminus\{0\}$ with $N(\alpha)=a$. Clearly, we have

[TABLE]

if and only if $\lambda x^{q^{i}}=\rho y^{q^{i}}$, for $i=0,\ldots,m-1$. If we compare the equalities with $i=0$ and $i=1$, we get $x^{q-1}=y^{q-1}$. For every fixed $x\in\mathbb{F}_{q^{m}}$ there are exactly $q-1$ elements $y$ in $\mathbb{F}_{q^{m}}$ such that $y^{q-1}=x^{q-1}$.

Let $\lambda$ and $x$ be fixed elements in $\mathbb{F}_{q^{m}}\setminus\{0\}$. Then, for each element $y\in\mathbb{F}_{q^{m}}$ such that $y^{q-1}=x^{q-1}$ we get the unique element $\rho=\lambda xy^{-1}$ and the result is proved. $\Box$

Lemma 2.10**.**

• i)

If $\mathbf{a}\in\pi_{1}$, then ${\rm rk}(f_{\mathbf{a}})=1$.

• ii)

If $a,b\in\mathbb{F}_{q}\setminus\{0,1\}$, then ${\rm rk}(f_{\mathbf{a}}-f_{\mathbf{b}})\geq m-1$, for any $\mathbf{a}\in\pi_{a}$ and $\mathbf{b}\in\pi_{b}$, with $\mathbf{b}\neq\mathbf{a}$ if $a=b$.

Proof.

i) Let $\mathbf{a}=(\lambda x,\lambda x^{q},\ldots,\lambda x^{q^{m-1}})\in\pi_{1}$. It suffices to note that $L_{\mathbf{a}}(z)=(\lambda x)z+(\lambda x^{q})z^{q}+\ldots(\lambda x^{q^{m-1}})z^{q^{m-1}}=0$ is the equation of a hyperplane in the cyclic model of $V$.

ii) By Remark 2.7, we can assume $\mathbf{a}=(1,\alpha,\ldots,a^{1+\ldots+q^{m-2}})$, with $N(\alpha)=a\neq 1$.

Let $\mathbf{b}=(\lambda x,\lambda\beta x^{q},\ldots,\lambda\beta^{1+q+\ldots+q^{m-2}}x^{q^{m-1}})$, with $N(\beta)=b\neq 1$.

Suppose there exist $z_{1},z_{2}\in\mathbb{F}_{q^{m}}$ linearly independent over $\mathbb{F}_{q}$ such that $L_{\mathbf{a}-\mathbf{b}}(z_{i})=0$. Then we get

[TABLE]

and

[TABLE]

for $i=1,2$.

After subtracting Equation (5) side-by-side from Equation (6) multiplied by $\alpha$, we get

[TABLE]

for $i=1,2$. Then, the $m-$ple

[TABLE]

is a solution of the linear system

[TABLE]

with $\Delta=\begin{vmatrix}z_{1}&z_{1}^{q}\\ z_{2}&z_{2}^{q}\end{vmatrix}\neq 0$.

The generic solution $(x_{1},x_{2},\ldots,x_{m})$ of (9) has

[TABLE]

and

[TABLE]

In the expression (11) set

[TABLE]

in particular $c_{m}=\begin{vmatrix}z_{1}&z_{1}^{q^{m-1}}\\ z_{2}&z_{2}^{q^{m-1}}\end{vmatrix}$ giving $c_{m}^{q}=-\Delta$. Similarly, in the expression (10) set

[TABLE]

We have,

[TABLE]

and $d_{3}=-\Delta^{q}$. We then write

[TABLE]

By plugging (8) in the right-hands of the above equalities we get

[TABLE]

and

[TABLE]

Therefore

[TABLE]

From (8), we have $x_{2}=(\lambda\beta-\lambda^{q}\alpha)x^{q}$ giving

[TABLE]

From (12) it turns out that the value of $x_{1}$ must satisfy

[TABLE]

giving

[TABLE]

since $\Delta\neq 0$.

From (8), we have $x_{1}=(a-1)+(\lambda-\lambda^{q}\alpha\beta^{q+\ldots+q^{m-1}})x$. Therefore, we get

[TABLE]

i.e.,

[TABLE]

By plugging this value in $\mathbf{b}$, we get

[TABLE]

Note that if $b=a$, we can assume $\beta=\alpha$ giving $x\not\in\mathbb{F}_{q}$ as $\mathbf{b}\neq\mathbf{a}$.

We claim that the bilinear form $(f_{\mathbf{a}}-f_{\mathbf{b}})$ has maximum rank $m$. Indeed, suppose there exists a nonzero $z\in\mathbb{F}_{q^{m}}$ such that $L_{\mathbf{a}-\mathbf{b}}(z)=0$. By plugging (13) in Equation (7) we get

[TABLE]

or, equivalently,

[TABLE]

where $\dfrac{\beta}{x}-\dfrac{\alpha}{x^{q}}\neq 0$ since either $b\neq a$ or $x^{q}\neq x$ if $b=a$. Therefore, the following equation holds:

[TABLE]

given

[TABLE]

By subtracting Equation (14) from (15) multiplied by $\beta^{q}$ we get $b=1$, a contradiction. $\Box$

For every nonzero element $\alpha\in\mathbb{F}_{q^{m}}$, let

[TABLE]

Remark 2.11**.**

Note that the set $\mathcal{F}_{J_{\alpha}}$ is the $(S\times S)$-orbit of the bilinear form $f_{\mathbf{a}}$, with $\mathbf{a}=(1,0,\ldots,0,-\alpha)$. It turns out that the bilinear forms in $\mathcal{F}_{J_{\alpha}}$ have constant rank.

By arguing similarly to the proof of Proposition 2.8 and Lemma 2.9, we get the following result.

Lemma 2.12**.**

Each set $J_{\alpha}$ has size $(q^{m}-1)^{2}/(q-1)$ and $J_{\alpha}=J_{\beta}$ if and only if $N(\alpha)=N(\beta)$.

We will write $J_{a}$ instead of $J_{\alpha}$, if $\alpha$ is an element of $\mathbb{F}_{q^{m}}$ with $N(\alpha)=a$.

Lemma 2.13**.**

For any $\mathbf{a}=(x,0,\ldots,0,y)$ with $x,y\in\mathbb{F}_{q^{m}}$ not both zero, ${\rm rk}(f_{\mathbf{a}})\geq m-1$.

Proof.

The bilinear form $f_{\mathbf{a}}$, is equivalent to the bilinear form $f_{\hat{\mathbf{a}}}$, with $\hat{\mathbf{a}}=(x,y^{q},0,\ldots,0)$, via the automorphism $\top$. The result then follows from Remark 2.5 and Theorem 6.3 in [11]. $\Box$

Corollary 2.14**.**

Let $a,b$ be nonzero elements in $\mathbb{F}_{q}$. Then ${\rm rk}(f_{\mathbf{a}}-f_{\mathbf{b}})\geq m-1$, for any $\mathbf{a}\in J_{a}$ and $\mathbf{b}\in J_{b}$, with $\mathbf{a}\neq\mathbf{b}$ if a=b.

Lemma 2.15**.**

Let $a,b$ be distinct nonzero elements in $\mathbb{F}_{q}$. Then ${\rm rk}(f_{\mathbf{a}}-f_{\mathbf{b}})\geq m-1$ for any $\mathbf{a}\in\pi_{a}$ and $\mathbf{b}\in J_{b}$.

Proof.

By Remark 2.7 we can assume $\mathbf{a}=(1,\alpha,\ldots,\alpha^{1+\ldots+q^{m-2}})$ with $N(\alpha)=a$. By arguing as in the proof of Lemma 2.10 we see that the triple

[TABLE]

is a solution of the linear system

[TABLE]

for some $z_{1},z_{2}\in\mathbb{F}_{q^{m}}$ linearly independent over $\mathbb{F}_{q}$ with $\Delta=\begin{vmatrix}z_{1}&z_{1}^{q}\\ z_{2}&z_{2}^{q}\end{vmatrix}\neq 0$. Any solution $(x_{1},x_{2},x_{3})$ of (17) satisfies

[TABLE]

where $\Delta^{\prime}=\begin{vmatrix}z_{1}&z_{1}^{q^{m-1}}\\ z_{2}&z_{2}^{q^{m-1}}\end{vmatrix}$. Since ${\Delta^{\prime}}^{q}=\begin{vmatrix}z_{1}^{q}&z_{1}\\ z_{2}^{q}&z_{2}\end{vmatrix}=-\Delta$ we get $x_{2}=\dfrac{1}{{\Delta^{\prime}}^{q-1}}x_{3}$ giving $N(x_{2})=N(x_{3})$. As a solution of (17), the triple (16) must satisfies $aN(\lambda)N(x)=bN(\lambda)N(x)$ giving either $\lambda x=0$ or $a=b$, a contradiction. $\Box$

Let $A_{1}=\{(x,0,0,\ldots,0):x\in\mathbb{F}_{q^{m}}\setminus\{0\}\}$ and $A_{2}=\{(0,0,0,\ldots,x):x\in\mathbb{F}_{q^{m}}\setminus\{0\}\}$.

Lemma 2.16**.**

${\rm rk}(f_{\mathbf{a}})=m$, for any $\mathbf{a}\in A_{i}$, $i=1,2$. Further, ${\rm rk}(f_{\mathbf{a}}-f_{\mathbf{b}})\geq m-1$, for any $\mathbf{a}\in A_{1}$ and $\mathbf{b}\in A_{2}$.

Proof.

The first part can be easily proved by taking the Dickson matrix $D_{\mathbf{a}}$ with $\mathbf{a}\in A_{i}$. The second part follows from Lemma 2.13. $\Box$

Lemma 2.17**.**

Let $a\in\mathbb{F}_{q}\setminus\{0,1\}$. Then ${\rm rk}(f_{\mathbf{a}}-f_{\mathbf{b}})\geq m-1$, for any $\mathbf{a}\in\pi_{a}$ and $\mathbf{b}\in A_{i}$, $i=1,2$.

Proof.

By Remark 2.7 we can assume $\mathbf{a}=(1,\alpha,\ldots,\alpha^{1+\ldots+q^{m-2}})$ with $N(\alpha)=a$. Let $\mathbf{b}=(x,0,\ldots,0)$. By proceeding as in the proof of Lemma 2.10 we see the pair $(a-(1-x),-\alpha x^{q})$ is a solution of the linear system

[TABLE]

with $\Delta=\begin{vmatrix}z_{1}&z_{1}^{q}\\ z_{2}&z_{2}^{q}\end{vmatrix}\neq 0$. Then the above linear system has the unique solution $(0,0)$ giving $x=0$ and $a=1$, a contradiction.

For $i=2$, similar arguments lead to the same contradiction. $\Box$

Lemma 2.18**.**

Let $a\in\mathbb{F}_{q}\setminus\{0\}$. Then ${\rm rk}(f_{\mathbf{a}}-f_{\mathbf{b}})\geq m-1$, for any $\mathbf{a}\in J_{a}$ and $\mathbf{b}\in A_{i}$, $i=1,2$.

Proof.

Use Lemma 2.13. $\Box$

Finally, we have the main theorem.

Theorem 2.19**.**

Let $q>2$ be a prime power and $m\geq 3$ a positive integer. For any subset $I$ of $\mathbb{F}_{q}\setminus\{0,1\}$, put $\Pi_{I}=\bigcup_{a\in I}{\pi_{a}}$, $\Gamma_{I}=\bigcup_{b\in\mathbb{F}_{q}\setminus(I\cup\{0\})}{J_{b}}$ and set

[TABLE]

where $\mathbf{0}$ is the zero $m-$ple. Then the subset $\mathcal{F}_{m,q;I}=\{f_{\mathbf{a}}:\mathbf{a}\in\mathcal{A}_{m,q;I}\}$ of $\Omega$ is a non-linear $(m,m,q;m-2)$-MRD code.

Proof.

By Lemmas 2.9, 2.12 we get that $\mathcal{A}_{m,q;I}$ has size $q^{2m}$. By Lemmas 2.10, 2.13, 2.15, 2.16, 2.17 and Corollary 2.14, we see that $\mathcal{F}_{m,q;I}$ has minimum distance $m-1$, i.e. it is a $(m,m,q;m-2)$-MRD code. To show the non-linearity of $\mathcal{F}_{m,q;I}$, it suffices to find two distinct elements in it whose $\mathbb{F}_{q}$-span is not contained in $\mathcal{F}_{m,q;I}$.

Let $f_{\mathbf{a}}\in\mathcal{F}_{A_{2}}$ and $f_{\mathbf{b}}\in\mathcal{F}_{\pi_{a}}$, $a\in I$. By corollary 2.3, we can work with the Dickson matrices $D_{\mathbf{a}}$ and $D_{\mathbf{b}}$, or equivalently, with $m$-ples $\mathbf{a}$ and $\mathbf{b}$ as arrays in $V(m,q^{m})$. Let $\mathbf{a}=(0,\ldots,0,\mu)$ and $\mathbf{b}=(\lambda x,\lambda\alpha x^{q},\ldots,\lambda\alpha^{1+\ldots+q^{m-2}}x^{q^{m-1}})$. Suppose $\mathbf{a}+\mathbf{b}\in\pi_{b}$, for some $b\in\mathbb{F}_{q}$. Then

[TABLE]

giving $\mu=0$. Therefore, the subspace spanned by $\mathbf{a}$ and $\mathbf{b}$ meets trivially every $\pi_{b}$ if $b\neq a$, or just in the 1-dimensional subspace spanned by $\mathbf{b}$ if $b=a$. The result then follows. $\Box$

3 A geometric description for the non-linear MRD codes

For any $v\in V(t,q^{s})\setminus\{0\}$, $[v]$ will denote the point of ${\rm PG}(t-1,q^{s})$ defined by $v$ via the canonical homomorphism $\psi:{\rm GL}(V(t,q^{s}))\mapsto{\rm PGL}(t,q^{s})$. For any subset $X$ of $V(t,q^{s})\setminus\{0\}$, we set $[X]=\{[v]:v\in X,v\neq 0\}$. The set $[X]$ is said to be an $\mathbb{F}_{q}$-linear set of rank $r$ if $X$ is an $r$-dimensional $\mathbb{F}_{q}$-linear subspace of $V(t,q^{s})$. An $\mathbb{F}_{q}$-linear set $[X]$ of rank $r$ is said to be scattered if the size of $[X]$ equals $|{\rm PG}(r-1,q)|$; see [31] for more details on $\mathbb{F}_{q}$-linear sets and [27] for a relationship between linear MRD-codes and $\mathbb{F}_{q}$-linear sets.

Consider the set $\mathcal{A}_{m,q;I}$ defined in Theorem 2.19 as a subset of $\widehat{V}=V(m,q^{m})$, by setting $a_{0}v_{1}+a_{1}v_{2}+\ldots+a_{m-1}v_{m}$, for any $\mathbf{a}=(a_{0},\ldots,a_{m-1})\in\mathcal{A}_{m,q;I}$; here, $v_{1},\ldots,v_{m}$ is the Singer basis of $V$ defined in Section 2. Therefore, $[\pi_{1}]=[V]$ is a scattered $\mathbb{F}_{q}$-linear set of rank $m$ of ${\rm PG}(m-1,q^{m})$ isomorphic to the projective space ${\rm PG}(m-1,q)$.

For any $\alpha\in\mathbb{F}_{q^{m}}\setminus\{0\}$, the endomorphism

[TABLE]

maps $\pi_{1}$ into $\pi_{a}$, with $a=N(\alpha)$, and $J_{1}$ into $J_{b}$, with $b=a^{m-1}$.

Let $W$ be the span of $v_{1}$ and $v_{m}$ in $\widehat{V}$. For any $a\in\mathbb{F}_{q}\setminus\{0\}$, $[J_{a}]$ is a scattered $\mathbb{F}_{q}$-linear set of rank $m$ of $[W]$. In particular $[J_{a}]$ is a maximum scattered $\mathbb{F}_{q}$-linear set of pseudoregulus type of $[W]$ [24, 29].

Summarizing we have the following result.

Theorem 3.1**.**

Let $q>2$ be a prime power and $m>2$ a positive integer. Let $I$ be any nonempty subset of $\mathbb{F}_{q}\setminus\{0,1\}$ with $k=|I|$. Then, the projective image of $\mathcal{A}_{m,q;I}$ in ${\rm PG}(m-1,q^{m})$ is union of two points $[A_{1}],[A_{2}]$, $k$ mutually disjoint $(m-1)$-dimensional $\mathbb{F}_{q}$-subgeometries $[\pi_{a}]$, $a\in I$, and $q-1-k$ mutually disjoint $\mathbb{F}_{q}$-linear sets $[J_{b}],b\in\mathbb{F}_{q}\setminus(I\cup\{0\})$, of pseudoregulus type of rank $m$ contained in the line spanned by $[A_{1}]$ and $[A_{2}]$.

We now investigate the geometry in ${\rm PG}(m^{2}-1,q)$ of the projective set defined by each MRD code $\mathcal{F}_{m,q;I}$ viewed as a subset of $V(m^{2},q)$.

Let $V=V(m,q)$ be the $\mathbb{F}_{q}$-span of $u_{1},\ldots,u_{m}$ and set $\widehat{V}=V(m,q^{m})=\mathbb{F}_{q^{m}}\otimes V(m,q)$. The rank of a vector $v=a_{1}u_{1}+a_{2}u_{2}+\ldots+a_{m}u_{m}\in\widehat{V}$ by definition is the maximum number of linearly independent coordinates $a_{i}$ over $\mathbb{F}_{q}$.

If we consider $\mathbb{F}_{q^{m}}$ as the $m$-dimensional vector space $V$, then every $\alpha\in\mathbb{F}_{q^{m}}$ can be uniquely written as $\alpha=x_{1}u_{1}+x_{2}u_{2}+\ldots+x_{m}u_{m}$, with $x_{i}\in\mathbb{F}_{q}$. Hence, $\widehat{V}$ can be viewed as $V\otimes V$, the tensor product of $V$ with itself, with basis $\{u_{(i,j)}=u_{i}\otimes u_{j}:i,j=1,\ldots,m\}$. Elements of $V\otimes V$ are called tensors and those of the form $v\otimes v^{\prime}$, with $v,v^{\prime}\in V$ are called fundamental tensors. In ${\rm PG}(V\otimes V)$, the set of fundamental tensors correspond to the Segre variety $\mathcal{S}_{m,m}(\mathbb{F}_{q})$ of ${\rm PG}(V\otimes V)$ [20].

Let $\phi$ be the map defined by

[TABLE]

with $\alpha_{k}=x_{1k}u_{1}+x_{2k}u_{2}+\ldots+x_{mk}u_{m}$, $x_{ik}\in\mathbb{F}_{q}$. We call this map the *field reduction of * $\widehat{V}$ over $\mathbb{F}_{q}$ with respect to the basis $u_{1},\ldots,u_{m}$. The projective space ${\rm PG}(V\otimes V)$ is the the field reduction of ${\rm PG}(\widehat{V})$ over $\mathbb{F}_{q}$ with respect to the basis $u_{1},\ldots,u_{m}$.

Under the map $\phi$, every 1-dimensional subspace $\langle v\rangle$ of $\widehat{V}$ is mapped to the $m$-dimensional $\mathbb{F}_{q}$-subspace $k_{v}=\phi(\langle v\rangle)$ of $V\otimes V$. It turns out that the set $\mathcal{K}=\{k_{v}:v\in\widehat{V},v\neq 0\}$ is a partition of the nonzero vectors of $V\otimes V$. In particular $\mathcal{K}$ is a Desarguesian partition, i.e. the stabilizer of $\mathcal{K}$ in ${\rm GL}(V\otimes V)$ contains a cyclic subgroup acting regularly on the components of $\mathcal{K}$ [34], [14].

To any component $k_{v}$ of $\mathcal{K}$ there corresponds a projective $(m-1)-$dimensional subspace $[k_{v}]$ of ${\rm PG}(V\otimes V)$. The set $\mathcal{S}=\{[k_{v}]:v\in\widehat{V},v\neq 0\}$ is so called a Desarguesian $(m-1)-$spread of ${\rm PG}(V\otimes V)$ [34], [14].

In addition, the projective set of ${\rm PG}(V\otimes V)$ corresponding to the $\phi$-image of the 1-dimensional subspaces spanned by non-zero vectors in $V$ is the Segre variety $\mathcal{S}_{m,m}(\mathbb{F}_{q})$.

Let $\nu$ be the map defined by

[TABLE]

For every $v=\alpha_{1}u_{1}+\ldots+\alpha_{m}u_{m}\in\widehat{V}$, the $k$-th column of the matrix $\nu(\phi(v))$ is the $m$-ple $(x_{1k},\ldots,x_{mk})$ of the coordinates of $\alpha_{k}$ with respect to the basis $u_{1},\ldots,u_{m}$ of $\mathbb{F}_{q^{m}}$. From [16], the rank of $v$ equals the rank of $\nu(\phi(v))$, for all $v\in\widehat{V}$. In addition, the $\nu$-image of fundamental tensors is precisely the set of rank 1 matrices.

Remark 3.2**.**

Evidently, $\nu$ is an isomorphism of rank metric spaces which also provides an isomorphism between the field reduction $V\otimes V$ of $\widehat{V}$ with respect to $u_{1},\ldots,u_{m}$ and the metric space $\Omega$ of all bilinear forms on $V=\langle u_{1},\ldots,u_{m}\rangle_{\mathbb{F}_{q}}$.

Now embed $V\otimes V$ into $\widehat{V}\otimes\widehat{V}$ by extending the scalars from $\mathbb{F}_{q}$ to $\mathbb{F}_{q^{m}}$. By taking a Singer basis $v_{1},\ldots,v_{m}$ of $V$ defined by the Singer cycle $\sigma$, Cooperstein [7] defined a cyclic model for $V\otimes V$ within $\widehat{V}\otimes\widehat{V}$ with basis $v_{(i,j)}=v_{i}\otimes v_{j}$, $i,j=1,\ldots,m$. Let

[TABLE]

where the subscript $j-1+i$ is taken modulo $m$. As an $\mathbb{F}_{q}$-space, $\Phi(j)$ has dimension $m$ and by consideration on dimension we have

[TABLE]

see [7]. We call this representation the cyclic representation of the tensor product $V\otimes V$.

Proposition 3.3**.**

Let $\widetilde{\phi}$ be the map defined by

[TABLE]

Then ${\rm Im}(\widetilde{\phi})$ is linearly equivalent to ${\rm Im}(\phi)$ in $\widehat{V}\otimes\widehat{V}$.

Proof.

Let $v=\sum_{i=1}^{m}{\alpha_{i}v_{i}}\in\widehat{V}$ be linear combination of $k$ vectors of rank 1, $1\leq k\leq m$.

Let $\tau$ be the change of basis map of $\widehat{V}$ from the basis $u_{1},\ldots,u_{m}$ to the Singer basis $v_{1},\ldots,v_{m}$.

Assume $k=1$, i.e. $v=\lambda(\sum_{i=1}^{m}{a^{q^{i-1}}v_{i}})$, and set $\lambda=\sum_{i=1}^{m}{l_{i}u_{i}}$, $a=\sum_{i=1}^{m}{x_{i}u_{i}}$, with $l_{i},x_{i}\in\mathbb{F}_{q}$. Therefore, $v=\lambda\sum_{i=1}^{m}{x_{i}u_{i}}$ and

[TABLE]

Now assume $v=\lambda_{1}(\sum_{i=1}^{m}{a_{1}^{q^{i-1}}v_{i}})+\ldots+\lambda_{k}(\sum_{i=1}^{m}{a_{k}^{q^{i-1}}v_{i}})$, $k>1$. Set $\lambda_{j}=\sum_{i=1}^{m}{l_{ij}u_{i}}$, $a_{j}=\sum_{i=1}^{m}{x_{ij}u_{i}}$, with $l_{ij},x_{ij}\in\mathbb{F}_{q}$. Therefore,

[TABLE]

giving $\phi(v)=\sum_{i=1}^{m}{(l_{i1}x_{11}+\ldots+l_{ik}x_{1k})u_{(i,1)}}+\ldots+\sum_{i=1}^{m}{(l_{i1}x_{m1}+\ldots+l_{ik}x_{mk})u_{(i,m)}}$.

On the other hand we have

[TABLE]

$\Box$

We call the map $\widetilde{\phi}$ the *field reduction of * $\widehat{V}$ over $\mathbb{F}_{q}$ with respect to the Singer basis $v_{1},\ldots,v_{m}$ and its image the cyclic model for the field reduction of $\widehat{V}$ over $\mathbb{F}_{q}$. The projective space whose points are the $1$-dimensional $\mathbb{F}_{q}-$subspaces generated by the elements of $\widetilde{\phi}(\widehat{V})$ is the cyclic model for the field reduction of ${\rm PG}(\widehat{V})$ over $\mathbb{F}_{q}$.

Let $\widetilde{\nu}$ be the map defined by

[TABLE]

Then, for any $v=\alpha_{1}v_{1}+\ldots+\alpha_{m}v_{m}\in\widehat{V}$, the matrix $\widetilde{\nu}(\widetilde{\phi}(v))$ is the Dickson matrix $D_{(\alpha_{1},\ldots,\alpha_{m})}$. Since the cyclic model for the field reduction of $\widehat{V}$ is obtained from the field reduction $\phi(\widehat{V})$ by changing a basis in $\widehat{V}\otimes\widehat{V}$, we get that the rank of $\widetilde{\nu}(\widetilde{\phi}(v))$ equals the rank of $\nu(\phi(v))$, for any $v\in\widehat{V}$.

In addition, the element $k_{v}=\widetilde{\phi}(\langle v\rangle)$ of the $m$-partition $\mathcal{K}$ is

[TABLE]

In particular, $\bigcup_{v\in V\setminus\{0\}}{\widetilde{\nu}(k_{v})}$ is the set of all rank 1 matrices in $\mathcal{D}_{m}(\mathbb{F}_{q^{m}})$.

From the arguments above, we see that the set $\mathcal{F}_{m,q;I}$ can be considered, via the isomorphism (3), as the field reduction of the set $\mathcal{A}_{m,q;I}$ with respect to the Singer basis $v_{1},\ldots,v_{m}$.

As $[\pi_{1}]=[V]$, then the set $\mathcal{F}_{\pi_{1}}=\widetilde{\phi}(\pi_{1})$ defines the Segre variety $\mathcal{S}_{m,m}(\mathbb{F}_{q})$ of ${\rm PG}(V\otimes V)$ and $\mathcal{F}_{\pi_{a}}$ defines a Segre variety projectively equivalent to $\mathcal{S}_{m,m}(\mathbb{F}_{q})$ under the element of ${\rm PGL}(V\otimes V)$ corresponding to the linear transformation $\tau_{\alpha}$ with $N(\alpha)=a$.

Remark 3.4**.**

Note that, whenever $a\neq 1$, elements in $\mathcal{F}_{\pi_{a}}$ have rank bigger then 1 by Lemma 2.10. This is explained by the fact that the linear transformation of $V\otimes V=V(m^{2},q)$ corresponding to $\tau_{\alpha}$ is not in ${\rm Aut}_{\mathbb{F}_{q}}(V\otimes V)$.

Let $W=\langle v_{1},v_{m}\rangle\subset\widehat{V}$. Then $\widetilde{\phi}(W)$ is a $2m$-dimensional vector subspace of $V\otimes V$. In $[\widetilde{\phi}(W)]$, the set $[\widetilde{\phi}(J_{1})]$ is the Bruck norm-surface

[TABLE]

introduced in [3] and widely investigated in [4, 5] and recently in [10, 23]. For any $x\in\mathbb{F}_{q^{m}}\setminus\{0\}$ set $J_{x}=\{\lambda xv_{1}-\lambda x^{q^{m-1}}v_{m}:\lambda\in\mathbb{F}_{q^{m}}\}$. Then $[\widetilde{\phi}(J_{x})]\subset\mathcal{N}$ and the set $\{[\widetilde{\phi}(J_{x})]:x\in\mathbb{F}_{q^{m}}\}$ is a so-called hyper-regulus of ${\rm PG}(\widetilde{W})$ [30]. It turns out, that under the linear transformation $\tau_{\alpha}$ with $N(\alpha)=a$, also $J_{a}$ defines a hyper-regulus of $[\widetilde{\phi}(W)]$.

The following result, which summarizes all above arguments, gives a geometric description of the MRD codes $\mathcal{F}_{m,q;I}$.

Theorem 3.5**.**

Let $q>2$ be a prime power and $m>2$ a positive integer. Let $I$ be any nonempty subset of $\mathbb{F}_{q}\setminus\{0,1\}$ with $k=|I|$. The projective image of the MRD code $\mathcal{F}_{m,q;I}$ in ${\rm PG}(m^{2}-1,q)$ is a subset of a Desarguesian spread which is union of two spread elements, $k$ mutually disjoint Segre varieties $\mathcal{S}_{m,m}(\mathbb{F}_{q})$ and $q-1-k$ mutually disjoint hypereguli all contained in the $(2m-1)$-dimensional projective subspace generated by the two spread elements.

4 The Cossidente-Marino-Pavese non-linear MRD code

Recently, Cossidente, Marino and Pavese constructed non-linear $(3,3,q;1)$-MRD codesin a totally geometric setting [8, Theorem 3.6].

In ${\rm PG}(2,q^{3})$, $q\geq 3$, let $\mathcal{C}$ be the set of points whose coordinates satisfy the equation $X_{1}X_{2}^{q}-X_{3}^{q+1}=0$, that is a $C_{F}^{1}$-set of ${\rm PG}(2,q^{3})$ as introduced and studied in [13]. The set $\mathcal{C}$ is the projective image of a subset of $V(3,q^{3})$ which is the union of $A_{1}$, $A_{2}^{\prime}=\{(0,x,0):x\in\mathbb{F}_{q^{3}}\setminus\{0\}\}$ and the $q-1$ sets $\gamma_{a}=\{(\lambda,\lambda x^{q+1},\lambda x^{q}):\lambda,x\in\mathbb{F}_{q^{3}}\setminus\{0\},N(x)=a\}$, with $a$ a nonzero element of $\mathbb{F}_{q}$.

For any nonzero $a\in\mathbb{F}_{q}$, let $\alpha\in\mathbb{F}_{q^{3}}$ with $N(\alpha)=a$ and set $Z_{a}=\{(\lambda x,-\lambda\alpha x^{q},0):\lambda,x\in\mathbb{F}_{q^{3}}\setminus\{0\}\}$. Let $I$ be any non-empty subset of $\mathbb{F}_{q}\setminus\{0,1\}$ and put

[TABLE]

Up to an endomorphism of $V\otimes V$ viewed as the vector space $V(9,q)$, the image of set $\mathcal{A}^{\prime}(q;I)$ under $\nu\circ\phi$ is a non-linear $(3,3,q;1)$-MRD code [8, Proposition 3.8].

Lemma 4.1**.**

Let $\theta$ be the semilinear transformation of $V(3,q^{3})$ defined by

[TABLE]

with associated automorphism $x\mapsto x^{q^{2}}$. Then $\theta$ maps $\gamma_{a}$ into $\pi_{a^{-1}}$ and $Z_{a}$ into $J_{a^{-1}}$, for any nonzero element $a$ of $\mathbb{F}_{q}$.

Proof.

Every element $x\in\mathbb{F}_{q^{3}}$ with $N(x)=a$ can be written as $x=\alpha t^{q-1}$ for some $t\in\mathbb{F}_{q^{3}}$ and $\alpha$ a fixed element in $\mathbb{F}_{q^{3}}$ such that $N(\alpha)=a$. By straightforward calculations, we can write $\gamma_{a}=\{(\lambda x,\lambda\alpha^{q+1}x^{q},\lambda\alpha^{q}x^{q^{2}}):\lambda,x\in\mathbb{F}_{q^{3}}\}$. Then, we get $\theta(\gamma_{a})=\{(\lambda x,\lambda(\alpha^{-1})^{q^{2}}x^{q},\lambda(\alpha^{-1})^{(q^{2}+1)}x^{q^{2}}):\lambda,x\in\mathbb{F}_{q^{3}}\}=\pi_{a^{-1}}$ as $N(\alpha^{-q^{2}})=N(\alpha^{-1})=a^{-1}$.

The last part of the statement follows from straightforward calculations. $\Box$

Corollary 4.2**.**

Let $I$ be any non-empty subset $I$ of $\mathbb{F}_{q}\setminus\{0,1\}$ and put $I^{-1}=\{a^{-1}:a\in I\}$. Then, up to the endomorphism $\theta$ of $V(3,q^{3})$ and the changing of basis in $V(3,q^{3})\otimes V(3,q^{3})$ from $u_{(i,j)}$ to $v_{(i,j)}$, the Cossidente-Marino-Pavese family of non-linear MRD codes is the set $\mathcal{F}_{3,q,I^{-1}}$.

Let $L$ be any line of ${\rm PG}(2,q^{3})$ disjoint from a subgeometry ${\rm PG}(2,q)$. The set of points of $L$ that lie on some proper subspace spanned by points of ${\rm PG}(2,q)$ is called the exterior splash of ${\rm PG}(2,q)$ on $L$ [25].

Proposition 4.3**.**

[10]** The exterior splash of the subgeometry $[\pi_{a}]$ on the line $[W]$ is the set $[J_{b}]$ with $b=a^{m-1}$.

Proof.

First we note that $[W]$ is disjoint from $[\pi_{1}]$. The $\mathbb{F}_{q^{m}}$-span of some hyperplane in the cyclic model of $V$ is a hyperplane of $\widehat{V}$ with equation $\sum_{i=1}^{m}{\alpha^{q^{i-1}}X_{i}}=0$, for some nonzero$\alpha\in\mathbb{F}_{q^{m}}$. As the Singer cycle $\sigma$ acts on the hyperplanes of $V$ by mapping the hyperplane with equation $\sum_{i=1}^{m}{\alpha^{q^{i-1}}X^{q^{i-1}}}=0$ to the hyperplane with equation $\sum_{i=1}^{m}{(\mu\alpha)^{q^{i-1}}X^{q^{i-1}}}=0$, then $\sigma$ maps the hyperplane of $\widehat{V}$ with equation $\sum_{i=1}^{m}{\alpha^{q^{i-1}}X_{i}}=0$ into the hyperplane with equation $\sum_{i=1}^{m}{(\mu\alpha)^{q^{i-1}}X_{i}}=0$. Note that $\sigma$ fixes $W$.

The hyperplane $\sum_{i=1}^{m}{X_{i}}=0$ of $\widehat{V}$ meets $W$ in the $\mathbb{F}_{q^{m}}$-subspace spanned by $v_{1}-v_{m}$. By looking at the action of the Singer cyclic group $S=\langle\sigma\rangle$ on $W$, we see that the exterior splash of $[\pi_{1}]$ on $[W]$ is the set $[J_{1}]$. By using he map $\tau_{\alpha}$ defined above with $N(\alpha)=a$, we get the result. $\Box$

Remark 4.4**.**

Let $U$ be the $\mathbb{F}_{q^{m}}$-span of $v_{1}$ and $v_{2}$ in $\widehat{V}$. It is evident that the semilinear transformation $\theta$ maps the exterior splash of $[\gamma_{a}]$ on $[U]$ into the exterior splash of $[\pi_{a^{-1}}]$ on $[W]$.

The exterior splash of $[\gamma_{a}]$ on $[U]$ is

[TABLE]

In [8], the splash of $[\gamma_{a}]$ was erroneusly given as the set $[Z_{a}]$. Note that, $[Z_{a}]$ never coincides with $[\gamma_{a}]$, unless $a=1$.