On $q$-covering designs

TL;DR

This paper improves upper bounds on the minimum size of certain $q$-covering designs in projective spaces by constructing new designs, advancing the understanding of their minimal configurations.

Contribution

The paper introduces new constructions that lead to tighter upper bounds on the size of specific $q$-covering designs, extending previous results.

Findings

Improved upper bounds for $ ext{C}_q(2n, 3, 2)$, $n geq 4$

Enhanced bounds for $ ext{C}_q(3n + 8, 4, 2)$, $n geq 0$

Refined bounds for $ ext{C}_q(2n, 4, 3)$, $n geq 4$

Abstract

A $q$-covering design $\mathbb{C}_q(n, k, r)$, $k \ge r$, is a collection $\mathcal X$ of $(k-1)$-spaces of $\mathrm{PG}(n-1, q)$ such that every $(r-1)$-space of $\mathrm{PG}(n-1, q)$ is contained in at least one element of $\mathcal X$ . Let $\mathcal{C}_q(n, k, r)$ denote the minimum number of $(k-1)$-spaces in a $q$-covering design $\mathbb{C}_q(n, k, r)$. In this paper improved upper bounds on $\mathcal{C}_q(2n, 3, 2)$, $n \ge 4$, $\mathcal{C}_q(3n + 8, 4, 2)$, $n \ge 0$, and $\mathcal{C}_q(2n,4,3)$, $n \ge 4$, are presented. The results are achieved by constructing the related $q$-covering designs.

Equations

$$\mbox{{\em rank}}\left(\begin{array}[]{c|c}I_{3}&A_{1}\\ I_{3}&A_{2}\end{array}\right)\leq 4.$$

$$\mbox{{\em rank}}\left(\begin{array}[]{c|c}I_{3}&A_{1}\\ I_{3}&A_{2}\end{array}\right)=\mbox{{\em rank}}\left(\begin{array}[]{c|c}I_{3}&A_{1}\\ 0&A_{2}-A_{1}\end{array}\right)=3+\mbox{{\em rank}}\left(A_{2}-A_{1}\right)\geq 5,$$

$$\frac{\left({\theta }_{m+2,q}-{\theta }_{m-1,q}\right)\left({\theta }_{m+1,q}-{\theta }_{m-1,q}\right)}{q+1}=q^{2m}(q^2+q+1)$$

$$\begin{gathered} q^{4n-6}+q^2\left(q^{4n-10}+q^{4n-12}+\dots +q^{2n-2}+q^{2n-4}+q^{2n-5}\right) \\ =q^{4n-6}+q^{4n-8}+\dots +q^{2n}+q^{2n-2}+q^{2n-3} \end{gathered}$$

$$\begin{gathered} (q+1)\left(q^{4n-10}+q^{4n-12}+\dots +q^{2n-2}+q^{2n-4}+q^{2n-5}\right)+ \\ +(q+1)\left(\sum _{i=2}^{n-2}{q}^{4i-6}+q^{4i-8}+\dots +q^{2i}+q^{2i-2}+q^{2i-3}\right)-1 \\ =(q+1)\left(\sum _{i=2}^{n-1}{q}^{4i-6}+q^{4i-8}+\dots +q^{2i}+q^{2i-2}+q^{2i-3}\right)-1 \end{gathered}$$

$$\lceil \frac{{\theta }_{n-1,q^2}{\theta }_{2n-2,q}}{q^2+q+1}\rceil \le {\mathcal{C}}_q(2n,3,2)\le q^{2n-2}{\theta }_{n-2,q^2}+q^{2n-3}-1+\sum _{i=2}^{n-1}\left({\theta }_{4i-5,q}-{\theta }_{2i-4,q}+q^{2i-2}\right).$$

$$\begin{gathered} q^{6n+8}+q^3\left(q^{6n+2}+q^{6n-1}+\dots +q^{3n+5}+q^{3n+2}+2q^{3n+1}-q^{3n-1}\right) \\ =q^{6n+8}+q^{6n+5}+\dots +q^{3n+8}+q^{3n+5}+2q^{3n+4}-q^{3n+2} \end{gathered}$$

$$\begin{gathered} (q^2+q+1)\left(q^{6n+2}+q^{6n-1}+\dots +q^{3n+5}+q^{3n+2}+2q^{3n+1}-q^{3n-1}\right)+ \\ +(q^2+q+1)\left(\sum _{i=0}^{n-2}{q}^{6i+8}+q^{6i+5}+\dots +q^{3i+8}+q^{3i+5}+2q^{3i+4}-q^{3i+2}\right)+q(q+1)(2q+1) \\ =(q^2+q+1)\left(\sum _{i=0}^{n-1}{q}^{6i+8}+q^{6i+5}+\dots +q^{3i+8}+q^{3i+5}+2q^{3i+4}-q^{3i+2}\right)+q(q+1)(2q+1) \end{gathered}$$

$$\begin{gathered} \lceil \frac{{\theta }_{3n+7,q}{\theta }_{3n+6,q}}{(q+1)(q^2+1)(q^2+q+1)}\rceil \le {\mathcal{C}}_q(3n+8,4,2)\le q^{3n+5}{\theta }_{n+1,q^3}+\sum _{i=0}^{n-1}\left({\theta }_{6i+10,q}-{\theta }_{3i+4,q}\right)+ \\ +\sum _{i=0}^n\left(q^{3i+2}(2q^2-1)\right)+q(q+1)(2q+1). \end{gathered}$$

$$\mathcal{Z}=\bigcup _{{\ell }^{\prime }\text{mbox}\text{em}lineof{\Sigma }^{\prime }}{\mathcal{Z}}_{{\ell }^{\prime }}$$

$$|{\mathbb{S}}_{n+1}|=q^{6(n-1)}+(q^2+1)(q^2+q+1)|{\mathbb{S}}_n|-q(q+1{)}^2(q^2+1){\beta }_n+q^3(q^2+q+1){\alpha }_n.$$

$${\beta }_{n+1}=(q^2+q+1)|{\mathbb{S}}_n|-(q^3+q^2+q){\beta }_n-q^2{\alpha }_n.$$

$$\mathcal{V}=\bigcup _{\ell \text{mbox}\text{em}lineof\Gamma }{\mathcal{V}}_{\ell }.$$

$$\mathcal{W}=\bigcup _{P\in \Gamma }{\mathcal{W}}_P.$$

$$\begin{gathered} |{\mathbb{S}}_{n+1}|=q^{6(n-1)}+(q^2+1)(q^2+q+1)\left(|{\mathbb{S}}_n|-{\beta }_n-q({\beta }_n-{\alpha }_n)\right)+(q+1)(q^2+1)({\beta }_n-{\alpha }_n)+{\alpha }_n \\ =q^{6(n-1)}+(q^2+1)(q^2+q+1)|{\mathbb{S}}_n|-q(q+1{)}^2(q^2+1){\beta }_n+q^3(q^2+q+1){\alpha }_n. \end{gathered}$$

$$\left(\bigcup _{\ell \text{mbox}\text{em}lineof\sigma }{\mathcal{V}}_{\ell }\right)\cup \left(\bigcup _{P\in \sigma }{\mathcal{W}}_P\right)\cup \bar{\mathcal{W}}.$$

$$\begin{gathered} {\beta }_{n+1}=(q^2+q+1)\left(|{\mathbb{S}}_n|-{\beta }_n-q({\beta }_n-{\alpha }_n)\right)+(q^2+q+1)({\beta }_n-{\alpha }_n)+{\alpha }_n \\ =(q^2+q+1)|{\mathbb{S}}_n|-q(q^2+q+1){\beta }_n-q^2{\alpha }_n. \end{gathered}$$

$$q^{12}+q^2(q^4+1)(q^2+1)(q^2+q+1)+1\le {\mathcal{C}}_q(8,4,3)\le q^{12}+q^4(q^2+1{)}^2(q^2+q+1)+1$$

$$\begin{gathered} (q^4+1)(q^6+q^3+1)(q^8+q^6+q^4+q^2+1)\le {\mathcal{C}}_q(10,4,3)\le \\ {q}^{18}+q^4(q^2+1)(q^2+q+1)(q^8+q^6+q^4+q^3+q^2+1)+1 \end{gathered}$$

Full text

Keywords: $q$–covering design; MRD–codes.

††F. Pavese: Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy;   e-mail: francesco.pavese@poliba.it

††Mathematics Subject Classification (2010): Primary 51E20, 05B40; Secondary 05B25, 51A05.

1 Introduction

Let $q$ be any prime power, let ${\rm GF}(q)$ be the finite field with $q$ elements and let ${\rm PG}(n-1,q)$ be the $(n-1)$–dimensional projective space over ${\rm GF}(q)$. We will use the term $k$–space to denote a subspace of ${\rm PG}(n-1,q)$ of projective dimension $k$. Let $t\leq s$. A *blocking set * $\mathbb{B}$ is a set of $(t-1)$–spaces of ${\rm PG}(n-1,q)$ such that every $(s-1)$–space of ${\rm PG}(n-1,q)$ contains at least one element of $\mathbb{B}$. In the last fifty years the general problem of determining the smallest cardinality of a blocking set $\mathbb{B}$ has been studied by several authors (see [17, 4] and references therein) and in very few cases has been completely solved [5, 2, 3, 9, 18].

A blocking set $\mathbb{B}$ can be seen as a $q$–analog of a well known combinatorial design, called Turán design, see [11], [10]. Indeed, a blocking set $\mathbb{B}$ is also called a $q$–Turán design $\mathbb{T}_{q}(n,t,s)$. The dual structure of a $q$–Turán design $\mathbb{T}_{q}(n,t,s)$ is called $q$–covering design and it is denoted with $\mathbb{C}_{q}(n,n-t,n-s)$. In other words, a $q$–covering design $\mathbb{C}_{q}(n,k,r)$ is a collection $\mathcal{X}$ of $(k-1)$–spaces of ${\rm PG}(n-1,q)$ such that every $(r-1)$–space of ${\rm PG}(n-1,q)$ is contained in at least one element of $\mathcal{X}$. Let $\mathcal{C}_{q}(n,k,r)$ denote the minimum number of $(k-1)$–spaces in a $q$–covering design $\mathbb{C}_{q}(n,k,r)$. Lower and upper bounds on $\mathcal{C}_{q}(n,k,r)$ were considered in [11], [10]. Lower bounds are obtained by analytical methods, and upper bounds are obtained by explicit constructions of the related $q$–covering designs. A $q$–covering design $\mathbb{C}_{q}(n,k,r)$ which cover every $(r-1)$–space exactly once is called $q$–Steiner system.

The concept of $q$–covering design is of interest not only in projective geometry and design theory, but also in coding theory. Indeed, in recent years there has been an increasing interest in $q$–covering designs due to their connections with constant–dimension codes. In particular, a $q$–Steiner system is an optimal constant–dimension code (so far there is only one known example of $q$–Steiner system, i.e., the $2$–covering design $\mathbb{C}_{2}(13,3,2)$ of smallest possible size [6]). Observe that, as shown in the inspiring article by Koetter and Kschischang [16], constant–dimension codes can be used for error–correction in random linear network coding theory.

In this paper we discuss bounds on $q$–covering designs. In Section 3, based on the $q$–covering design $\mathbb{C}_{q}(6,3,2)$ constructed in [7], an improved upper bound on $\mathcal{C}_{q}(2n,3,2)$, $n\geq 4$, is presented. In the last two sections, starting from a lifted MRD–code, improvements on the upper bounds of $\mathcal{C}_{q}(3n+8,4,2)$, $n\geq 0$, and $\mathcal{C}_{q}(2n,4,3)$, $n\geq 4$, are obtained. In particular, first $q$–covering designs $\mathbb{C}_{q}(8,4,r)$, $r=2,3$, of ${\rm PG}(7,q)$ are constructed. Then, by induction, $q$–covering designs $\mathbb{C}_{q}(3n+8,4,2)$, $n\geq 0$, and $\mathbb{C}_{q}(2n,4,3)$, $n\geq 4$, are presented.

In the sequel we will use the following notation $\theta_{n,q}:=\genfrac{[}{]}{0.0pt}{}{n+1}{1}_{q}=q^{n}+\ldots+q+1$.

2 Preliminaries

A conic of ${\rm PG}(2,q)$ is the set of points of ${\rm PG}(2,q)$ satisfying a quadratic equation: $a_{11}X_{1}^{2}+a_{22}X_{2}^{2}+a_{33}X_{3}^{2}+a_{12}X_{1}X_{2}+a_{13}X_{1}X_{3}+a_{23}X_{2}X_{3}=0$. There exist four kinds of conics in ${\rm PG}(2,q)$, three of which are degenerate (splitting into lines, which could be in the plane ${\rm PG}(2,q^{2})$) and one of which is non–degenerate, see [13].

A regulus is the set of lines intersecting three skew (disjoint) lines and has size $q+1$. The hyperbolic quadric $\mathcal{Q}^{+}(3,q)$, is the set of points of ${\rm PG}(3,q)$ which satisfy the equation $X_{1}X_{2}+X_{3}X_{4}=0$. The hyperbolic quadric $\mathcal{Q}^{+}(3,q)$ consists of $(q+1)^{2}$ points and $2(q+1)$ lines that are the union of two reguli. Through a point of $\mathcal{Q}^{+}(3,q)$ there pass two lines belonging to different reguli.

A line–spread of ${\rm PG}(3,q)$ is a set $\mathcal{S}$ of $q^{2}+1$ lines of ${\rm PG}(3,q)$ with the property that each point of ${\rm PG}(3,q)$ is incident with exactly one element of $\mathcal{S}$. A $1$–parallelism of ${\rm PG}(3,q)$ is a collection $\mathcal{P}$ of $q^{2}+q+1$ line–spreads such that each line of ${\rm PG}(3,q)$ is contained in exactly one line–spread of $\mathcal{P}$. In [1] the author proved that there exist $1$–parallelisms in ${\rm PG}(3,q)$.

The Klein quadric $\mathcal{Q}^{+}(5,q)$, is the set of points of ${\rm PG}(5,q)$ which satisfy the equation $X_{1}X_{2}+X_{3}X_{4}+X_{5}X_{6}=0$. The Klein quadric contains $(q^{2}+1)(q^{2}+q+1)$ points of and two families each consisting of $q^{3}+q^{2}+q+1$ planes called Latin planes and Greek planes. Two distinct planes in the same family share exactly one point, whereas planes lying in distinct families are either disjoint or meet in a line. A line of ${\rm PG}(5,q)$ not contained in $\mathcal{Q}^{+}(5,q)$ is either external, or tangent, or secant to $\mathcal{Q}^{+}(5,q)$, according as it contains $0,1$ or $2$ points of $\mathcal{Q}^{+}(5,q)$. A hyperplane of ${\rm PG}(5,q)$ contains either $q^{3}+2q^{2}+q+1$ or $q^{3}+q^{2}+q+1$ points of $\mathcal{Q}^{+}(5,q)$. In the former case the hyperplane is called tangent, contains the $2(q+1)$ planes of $\mathcal{Q}^{+}(5,q)$ through one of its points, say $R$, and meets $\mathcal{Q}^{+}(5,q)$ in a cone having as vertex the point $R$ and as base a hyperbolic quadric $\mathcal{Q}^{+}(3,q)$. In the latter case the hyperplane is called secant and contains no plane of $\mathcal{Q}^{+}(5,q)$. The stabilizer of $\mathcal{Q}^{+}(5,q)$ in ${\rm PGL}(6,q)$, say $G$, contains a subgroup isomorphic to ${\rm PGL}(4,q)$. Also, the stabilizer in $G$ of a plane $g$ of $\mathcal{Q}^{+}(5,q)$ contains a subgroup $H$ isomorphic to ${\rm PGL}(3,q)$ acting in its natural representation on the points and lines of $g$. For more details see [14, Chapter 1]. A Singer cyclic subgroup of ${\rm PGL}(3,q)$ is a cyclic group acting regularly on points and lines of a projective plane ${\rm PG}(2,q)$.

2.1 Lifting an MRD–code

The set ${\cal M}_{n\times m}(q)$, $n\leq m$, of $n\times m$ matrices over the finite field ${\rm GF}(q)$ forms a metric space with respect to the rank distance defined by $d_{r}(A,B)=\mbox{{\em rank}}(A-B)$. The maximum size of a code of minimum distance $\delta$, with $1\leq\delta\leq n$, in $\left({\cal M}_{n\times m}(q),d_{r}\right)$ is $q^{n(m-\delta+1)}$. A code $\mathcal{A}\subset\mathcal{M}_{n\times m}(q)$ attaining this bound is said to be a $(n\times m,\delta)_{q}$ maximum rank distance code (or MRD–code in short). A rank distance code $\mathcal{A}$ is called ${\rm GF}(q)$–linear if $\cal A$ is a subspace of ${\cal M}_{n\times m}(q)$ considered as a vector space over ${\rm GF}(q)$. Linear MRD–codes exist for all possible parameters [8, 12, 19, 20].

We recall the so–called lifting process for a matrix $A\in\mathcal{M}_{n\times m}(q)$, see [21]. Let $I_{n}$ be the $n\times n$ identity matrix. The rows of the $n\times n+m$ matrix $(I_{n}|A)$ can be viewed as coordinates of points in general position of an $(n-1)$–space of ${\rm PG}(n+m-1,q)$. This subspace is denoted by $L(A)$. Hence the matrix $A$ can be “lifted” to the $(n-1)$–space $L(A)$. Let $U_{i}$ be the point of ${\rm PG}(n+m-1,q)$ having $1$ in $i$–th position and [math] elsewhere, $1\leq i\leq n+m$, and let $\Sigma$ be the $(m-1)$–space of ${\rm PG}(n+m-1,q)$ containing $U_{n+1},\ldots,U_{n+m}$. Note that if $A\in\mathcal{A}$, then $L(A)$ is disjoint from $\Sigma$.

Proposition 2.1**.**

• i)

If $\mathcal{A}$ is a $(3\times m,2)_{q}$ MRD–code, $m\geq 3$, then $\mathcal{X}=\{L(A)\;|\;A\in\mathcal{A}\}$ is a set of $q^{2m}$ planes of ${\rm PG}(m+2,q)$ such that every line of ${\rm PG}(m+2,q)$ disjoint from $\Sigma$ is contained in exactly one element of $\mathcal{X}$.

• ii)

If $\mathcal{A}$ is a $(4\times m,3)_{q}$ MRD–code, $m\geq 4$, then $\mathcal{X}=\{L(A)\;|\;A\in\mathcal{A}\}$ is a set of $q^{2m}$ solids of ${\rm PG}(m+3,q)$ such that every line of ${\rm PG}(m+3,q)$ disjoint from $\Sigma$ is contained in exactly one element of $\mathcal{X}$.

• iii)

If $\mathcal{A}$ is a $(4\times m,2)_{q}$ MRD–code, $m\geq 4$, then $\mathcal{X}=\{L(A)\;|\;A\in\mathcal{A}\}$ is a set of $q^{3m}$ solids of ${\rm PG}(m+3,q)$ such that every plane of ${\rm PG}(m+3,q)$ disjoint from $\Sigma$ is contained in exactly one element of $\mathcal{X}$.

Proof.

i)  Let $L(A_{1}),L(A_{2})\in\mathcal{X}$, where $A_{1},A_{2}\in\mathcal{A}$ and $A_{1}\neq A_{2}$. Assume by contradiction that $L(A_{1})\cap L(A_{2})$ contains a line. Then we would have

[TABLE]

On the other hand,

[TABLE]

a contradiction. Therefore the set $\mathcal{X}$ contains $q^{2m}$ planes pairwise meeting in at most a point. Hence there are $(q^{2}+q+1)q^{2m}$ lines of ${\rm PG}(m+2,q)$ lying on a plane of $\mathcal{X}$. But

[TABLE]

is the total number of lines of ${\rm PG}(n+m+1,q)$ that are disjoint from $\Sigma$.

A similar argument ban be used in the remaining cases i), ii). ∎

3 On $\mathcal{C}_{q}(2n,3,2)$

In this section we provide an upper bound on $\mathcal{C}_{q}(2n,3,2)$, $n\geq 4$. In [7], a constructive upper bound on $\mathcal{C}_{q}(6,3,2)$ has been given. In what follows we recall the construction and some of the properties of this $q$–covering design.

Construction 3.1**.**

Let $g$ be a Greek plane of $\mathcal{Q}^{+}(5,q)$. From [7, Lemma 2.2], there exists a set $\mathcal{X}$ of $q^{6}-q^{3}$ planes disjoint from $g$ and meeting $\mathcal{Q}^{+}(5,q)$ in a non–degenerate conic that, together with the set $\mathcal{Y}$ of $q^{3}+q^{2}+q$ Greek planes of $\mathcal{Q}^{+}(5,q)$ distinct from $g$, cover every line $\ell$ of ${\rm PG}(5,q)$ that is either disjoint from $g$ or contained in $\mathcal{Q}^{+}(5,q)\setminus g$.

Let $\ell$ be a line of $g$. Through the line $\ell$ there pass $q-1$ planes meeting $\mathcal{Q}^{+}(5,q)$ exactly in $\ell$ and a unique Latin plane $\pi$. Varying the line $\ell$ over the plane $g$ and considering the planes meeting $\mathcal{Q}^{+}(5,q)$ exactly in $\ell$, we get a family $\mathcal{Z}$ of consisting of $(q-1)(q^{2}+q+1)=q^{3}-1$ planes. From [7, Lemma 2.3], every line that is tangent to $\mathcal{Q}^{+}(5,q)$ at a point of $g$ is contained in exactly a plane of $\mathcal{Z}$.

Let $P$ be a point of $\ell$. Through the point $P$ there pass $q$ lines of $\pi$ and $q$ lines of $g$ distinct from $\ell$ and contained in $\mathcal{Q}^{+}(5,q)$. Let $S$ be the set of $q^{2}$ planes generated by a line of $\pi$ through $P$ distinct from $\ell$ and a line of $g$ through $P$ distinct from $\ell$. Let $C$ be a Singer cyclic group of the group $H\simeq{\rm PGL}(3,q)$. Here $H$ is a subgroup of $G$ stabilizing the plane $g$. Let $\mathcal{T}$ be the orbit of the set $S$ under $C$. Then $\mathcal{T}$ consists of $q^{2}(q^{2}+q+1)$ planes and each of these planes has $2q+1$ points in common with $\mathcal{Q}^{+}(5,q)$ on two intersecting lines of $\mathcal{Q}^{+}(5,q)$. From [7, Lemma 2.4], every line that is secant to $\mathcal{Q}^{+}(5,q)$ and has a point on $g$ is contained in exactly one plane of $\mathcal{T}$.

Theorem 3.2**.**

[7, Theorem 2.5]** The set $\mathcal{X}\cup\mathcal{Y}\cup\mathcal{Z}\cup\mathcal{T}$ is a $q$–covering design $\mathbb{C}_{q}(6,3,2)$ of size $q^{6}+q^{4}+2q^{3}+2q^{2}+q-1$.

We will need the following result.

Theorem 3.3**.**

There exists a hyperplane $\Gamma$ of ${\rm PG}(5,q)$ such that $q^{3}+2q^{2}+q-1$ elements of $\mathcal{X}\cup\mathcal{Y}\cup\mathcal{Z}\cup\mathcal{T}$ are contained in $\Gamma$.

Proof.

Let $\Gamma$ be a hyperplane of ${\rm PG}(5,q)$ containing $g$. Then $\Gamma$ is a tangent hyperplane and contains the planes of $\mathcal{Q}^{+}(5,q)$ through a points $R$ of $g$. In particular, there are $q$ planes of $\mathcal{Y}$ contained in $\Gamma$. First of all observe that no plane of $\mathcal{X}$ is contained in $\Gamma$. Indeed, by way of contradiction, assume that a plane of $\mathcal{X}$ is contained in $\Gamma$. Then such a plane would meet $g$ in at least a point, contradicting the fact that every plane of $\mathcal{X}$ is disjoint from $g$. A plane of $\mathcal{Z}$ that is contained in $\Gamma$ has to contain the point $R$. On the other hand, the $q-1$ planes of $\mathcal{Z}$, passing through a line of $g$ which is incident with $R$, are contained in $\Gamma$. Hence there are $(q+1)(q-1)=q^{2}-1$ planes of $\mathcal{Z}$ contained in $\Gamma$. If $\pi$ is a Latin plane contained in $\Gamma$, then $\pi\cap g$ is a line, say $\ell$. By construction there is a point $P\in\ell$ such that the set $\mathcal{T}$ contains $q^{2}$ planes meeting $\pi$ in a line through $P$ and $g$ in a line through $P$. Note that these $q^{2}$ planes of $\mathcal{T}$ are contained in $\Gamma$. It follows that there are $q^{2}(q+1)$ planes of $\mathcal{T}$ contained in $\Gamma$. The result follows. ∎

The $q$–covering design of Theorem 3.2 can be used recursively to obtain a $q$–covering design $\mathbb{C}_{q}(2n,3,2)$, $n\geq 4$, as described in the following result.

Theorem 3.4**.**

There exists a $q$–covering design $\mathbb{C}_{q}(2n,3,2)$, $n\geq 3$, and a hyperplane $\Gamma$ of ${\rm PG}(2n-1,q)$ such that there are $q^{2n-3}+\sum_{j=0}^{n-2}q^{2(n+j-1)}$ planes of $\mathbb{C}_{q}(2n,3,2)$ not contained in $\Gamma$ and $(q+1)\left(\sum_{i=2}^{n-1}\left(q^{2i-3}+\sum_{j=0}^{i-2}q^{2(i+j-1)}\right)\right)-1$ planes of $\mathbb{C}_{q}(2n,3,2)$ contained in $\Gamma$.

Proof.

By induction on $n$. If $n=3$, then, from Theorem 3.2 and Theorem 3.3, the result holds true. Assume that the result holds true for $n-1$: there are $q^{4n-10}+q^{4n-12}+\ldots+q^{2n-2}+q^{2n-4}+q^{2n-5}$ planes of a $\mathbb{C}_{q}(2n-2,3,2)$ not contained in a $(2n-4)$–space $\bar{\Lambda}$ of ${\rm PG}(2n-3,q)$ and $(q+1)\left(\sum_{i=2}^{n-2}q^{4i-6}+q^{4i-8}+\ldots+q^{2i}+q^{2i-2}+q^{2i-3}\right)-1$ planes of $\mathbb{C}_{q}(2n,3,2)$ contained in $\bar{\Lambda}$.

We claim that the result holds true for $n$. In ${\rm PG}(2n-1,q)$, let $\Lambda$ be the $(2n-4)$–space $\langle U_{4},\ldots,U_{2n}\rangle$. Let $\mathcal{A}$ be a $(3\times(2n-3),2)_{q}$ MRD–code and let $\mathcal{U}=\{L(A)\;|\;A\in\mathcal{A}\}$ be the set of $q^{4n-6}$ planes of ${\rm PG}(2n-1,q)$ obtained by lifting the matrices of $\mathcal{A}$. Let $\pi$ be a plane disjoint from $\Lambda$. For every point $P\in\pi$, let $\Lambda_{P}$ be the $(2n-3)$–space $\langle\Lambda,P\rangle$. From the induction hypothesis there is a $q$–covering design $\mathbb{C}_{q}(2n-2,3,2)$ of $\Lambda_{P}$, say $\mathcal{D}_{P}$, such that there is a subset of $\mathcal{D}_{P}$, say $\bar{\mathcal{D}}_{P}$, consisting of the $q^{4n-10}+q^{4n-12}+\ldots+q^{2n-2}+q^{2n-4}+q^{2n-5}$ planes of $\mathcal{D}_{P}$ not contained in $\Lambda$. Moreover $|\mathcal{D}_{P}\setminus\bar{\mathcal{D}}_{P}|=(q+1)\left(\sum_{i=2}^{n-2}q^{4i-6}+q^{4i-8}+\ldots+q^{2i}+q^{2i-2}+q^{2i-3}\right)-1$.

Let $\mathcal{V}=\bigcup_{P\in\pi}\bar{\mathcal{D}}_{P}$ and let $\mathcal{W}=\mathcal{D}_{P}\setminus\bar{\mathcal{D}}_{P}$. First of all observe that $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ is a $q$–covering design $\mathbb{C}_{q}(2n,3,2)$. Indeed, from Proposition 2.1, every line of ${\rm PG}(2n-1,q)$ disjoint from $\Lambda$ is contained in exactly one element of $\mathcal{U}$. On the other hand, if $r$ is a line of ${\rm PG}(2n-1,q)$ meeting $\Lambda$ in at least a point, then $r$ is contained in $\Lambda_{P}$, for some $P\in\pi$, and $r$ is contained in at least a plane of $\mathcal{D}_{P}$. Hence $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ is a $q$–covering design $\mathbb{C}_{q}(2n,3,2)$.

Let $\ell$ be a line of $\pi$ and let $\Gamma$ be the hyperplane $\langle\Lambda,\ell\rangle$ of ${\rm PG}(2n-1,q)$. We will prove that there are $q^{4n-6}+q^{4n-8}+\ldots+q^{2n}+q^{2n-2}+q^{2n-3}$ planes of $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ not contained in $\Gamma$. Since every plane of $\mathcal{U}$ is disjoint from $\Lambda$, we have that no plane of $\mathcal{U}$ is contained in $\Gamma$. The planes of $\mathcal{V}$ not contained in $\Gamma$ are those of the set $\bigcup_{P\in\pi,P\notin\ell}\bar{\mathcal{D}}_{P}$. Moreover the planes of $\mathcal{W}$ are contained in $\Gamma$. Hence there are

[TABLE]

planes of $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ not contained in $\Gamma$. Finally note that the planes of $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ contained in $\Gamma$ are those of $\bigcup_{P\in\ell}\bar{\mathcal{D}}_{P}\cup\mathcal{W}$. Hence there are

[TABLE]

planes of $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ contained in $\Gamma$. ∎

Theorem 3.4, together with [10, Theorem 1] gives the following result.

Corollary 3.5**.**

If $n\geq 3$, then

[TABLE]

4 On $\mathcal{C}_{q}(3n+8,4,2)$

In this section we provide an upper bound on $\mathcal{C}_{q}(3n+8,4,2)$, $n\geq 0$. We first deal with the case $n=0$.

Construction 4.1**.**

Let $\mathcal{A}$ be a $(4\times 4,3)_{q}$ MRD–code and let $\mathcal{X}=\{L(A)\;|\;A\in\mathcal{A}\}$ be the set of $q^{8}$ solids of ${\rm PG}(7,q)$ obtained by lifting the matrices of $\mathcal{A}$. Let $\Sigma^{\prime}$ be the solid of ${\rm PG}(7,q)$ containing $U_{1},U_{2},U_{3},U_{4}$. Then $\Sigma^{\prime}$ is disjoint from $\Sigma$. Let $\mathcal{S}=\{\ell_{i}\;|\;1\leq i\leq q^{2}+1\}$ be a line–spread of $\Sigma$, let $\mathcal{S}^{\prime}=\{\ell_{i}^{\prime}\;|\;1\leq i\leq q^{2}+1\}$ be a line–spread of $\Sigma^{\prime}$ and let $\mu:\ell_{i}^{\prime}\in\mathcal{S}^{\prime}\longmapsto\ell_{i}\in\mathcal{S}$ be a bijection. Let $\Gamma_{i}$ denote the $5$–space containing $\Sigma$ and $\ell_{i}^{\prime}$, $1\leq i\leq q^{2}+1$. If $\gamma$ is a plane of $\Gamma_{i}$, then there are $q^{2}+q$ solids of $\Gamma_{i}$ meeting $\Sigma_{i}$ exactly in $\gamma$. Let $\mathcal{Y}_{i}$ be the set of $q(q+1)^{2}$ solids of $\Gamma_{i}$ (distinct from $\Sigma$) meeting $\Sigma$ in a plane containing $\mu(\ell_{i}^{\prime})$. Let $\mathcal{Y}=\bigcup_{i=1}^{q^{2}+1}\mathcal{Y}_{i}$. Then $\mathcal{Y}$ consists of $q(q+1)^{2}(q^{2}+1)$ solids.

Theorem 4.2**.**

The set $\mathcal{X}\cup\mathcal{Y}$ is a $q$–covering design $\mathbb{C}_{q}(8,4,2)$ of size $q^{8}+q(q+1)^{2}(q^{2}+1)$.

Proof.

Let $r$ be a line of ${\rm PG}(7,q)$. If $r$ is disjoint from $\Sigma$, then from Proposition 2.1, we have that $r$ is contained in exactly one element of $\mathcal{X}$. If $r$ meets $\Sigma$ in one point, say $P$, then let $\Lambda$ be the $4$–space $\langle\Sigma,r\rangle$, let $\ell_{j}$ be the unique line of $\mathcal{S}$ containing $P$, let $P^{\prime}$ be the point $\Sigma^{\prime}\cap\Lambda$ and let $\ell_{k}^{\prime}$ be the unique line of $\mathcal{S}^{\prime}$ containing $P^{\prime}$. If $j=k$, then $P\in\ell_{k}$ and $r$ is contained in the $q+1$ solids $\langle\alpha,r\rangle$ of $\mathcal{Y}$, where $\alpha$ is a plane of $\Sigma$ containing $\ell_{k}$. If $j\neq k$, then $P\notin\ell_{k}$. Let $\beta$ be the plane of $\Sigma$ containing $\ell_{k}$ and $P$. Then $r$ is contained in $\langle\beta,r\rangle$, where $\langle\beta,r\rangle$ is a solid of $\mathcal{Y}$. Finally let $r$ be a line of $\Sigma$, then $r$ is contained in $q(q+1)^{2}$ solids of $\mathcal{Y}$. ∎

Remark 4.3**.**

Let $\mathcal{L}$ be a Desarguesian line–spread of ${\rm PG}(7,q)$. There are $(q^{4}+1)(q^{4}+q^{2}+1)$ solids of ${\rm PG}(7,q)$ containing exactly $q^{2}+1$ lines of $\mathcal{L}$. If $\mathcal{Z}$ denotes the set of these solids, then it is not difficult to see that every line of ${\rm PG}(7,q)$ is contained in at least a solid of $\mathcal{Z}$. In [17, p. 221], K. Metsch posed the following question: “Is $(q^{4}+1)(q^{4}+q^{2}+1)$ the smallest cardinality of a set of $3$–spaces of ${\rm PG}(7,q)$ that cover every line?” Theorem 4.2 provides a negative answer to this question.

Remark 4.4**.**

When $q=2$, the result of Theorem 4.2 was obtained in [10, Theorem 13].

Proposition 4.5**.**

There exists a hyperplane $\Gamma$ of ${\rm PG}(7,q)$ such that precisely $q(q+1)(2q+1)$ members of $\mathcal{X}\cup\mathcal{Y}$ are contained in $\Gamma$.

Proof.

Let $\Gamma$ be a hyperplane of ${\rm PG}(7,q)$ containing $\Sigma$. Then no element of $\mathcal{X}$ is contained in $\Gamma$, otherwise such a solid would meet $\Sigma$, contradicting the fact that every solid in $\mathcal{X}$ is disjoint from $\Sigma$. The hyperplane $\Gamma$ intersects $\Sigma^{\prime}$ in a plane $\sigma$. The plane $\sigma$ contains exactly one line of $\mathcal{S}^{\prime}$, say $\ell_{k}^{\prime}$. Hence the $q(q+1)^{2}$ solids of $\mathcal{Y}$ meeting $\Sigma$ in a plane through the line $\mu(\ell_{k}^{\prime})=\ell_{k}$ are contained in $\Gamma$. Let $\ell_{j}^{\prime}\in\mathcal{S}^{\prime}$, with $j\neq k$, then $\ell_{j}^{\prime}\cap\sigma$ is a point, say $R$. In this case the $q+1$ solids generated by $R$ and a plane of $\Sigma$ through $\mu(\ell_{j}^{\prime})=\ell_{j}$ is contained in $\Gamma$. Since the elements of $\mathcal{Y}$ are those contained in the $5$–space $\langle\Sigma,\ell_{i}^{\prime}\rangle$, where $\ell_{i}^{\prime}\in\mathcal{S}^{\prime}$, and meeting $\Sigma$ in a plane through $\ell_{i}$, the proof is complete. ∎

The $q$–covering design of Theorem 4.2 can be used recursively to obtain a $q$–covering design $\mathbb{C}_{q}(3n+8,4,2)$, $n\geq 1$, as described in the following result.

Theorem 4.6**.**

There exists a $q$–covering design $\mathbb{C}_{q}(3n+8,4,2)$, $n\geq 0$, and a hyperplane $\Gamma$ of ${\rm PG}(3n+7,q)$ such that there are $q^{3n+2}(2q^{2}-1)+\sum_{j=0}^{n+1}q^{3(n+j)+5}$ solids of $\mathbb{C}_{q}(3n+8,4,2)$ not contained in $\Gamma$ and $(q^{2}+q+1)\left(\sum_{i=0}^{n-1}\left(q^{3i+2}(2q^{2}-1)+\sum_{j=0}^{i+1}q^{3(i+j)+5}\right)\right)+q(q+1)(2q+1)$ solids of $\mathbb{C}_{q}(3n+8,4,2)$ contained in $\Gamma$.

Proof.

By induction on $n$. If $n=0$, then, from Theorem 4.2 and Proposition 4.5, the result holds true. Assume that the result holds true for $n-1$: there are $q^{6n+2}+q^{6n-1}+\ldots+q^{3n+5}+q^{3n+2}+2q^{3n+1}-q^{3n-1}$ solids of a $\mathbb{C}_{q}(3n+5,4,2)$ not contained in a $3(n+1)$–space $\bar{\Lambda}$ of ${\rm PG}(3n+4,q)$ and $(q^{2}+q+1)\left(\sum_{i=0}^{n-2}q^{6i+8}+q^{6i+5}+\ldots+q^{3i+8}+q^{3i+5}+2q^{3i+4}-q^{3i+2}\right)+q(q+1)(2q+1)$ solids of $\mathbb{C}_{q}(3n+5,4,2)$ contained in $\bar{\Lambda}$.

We claim that the result holds true for $n$. In ${\rm PG}(3n+7,q)$, let $\Lambda$ be the $3(n+1)$–space $\langle U_{5},\ldots,U_{3n+8}\rangle$. Let $\mathcal{A}$ be a $(4\times(3n+4),3)_{q}$ MRD–code and let $\mathcal{U}=\{L(A)\;|\;A\in\mathcal{A}\}$ be the set of $q^{6n+8}$ solids of ${\rm PG}(3n+7,q)$ obtained by lifting the matrices of $\mathcal{A}$. Let $\Pi$ be a solid disjoint from $\Lambda$. For every point $P\in\Pi$, let $\Lambda_{P}$ be the $(3n+4)$–space $\langle\Lambda,P\rangle$. From the induction hypothesis there is a $q$–covering design $\mathbb{C}_{q}(3n+5,4,2)$ of $\Lambda_{P}$, say $\mathcal{D}_{P}$, such that there is a subset of $\mathcal{D}_{P}$, say $\bar{\mathcal{D}}_{P}$, consisting of the $q^{6n+2}+q^{6n-1}+\ldots+q^{3n+5}+q^{3n+2}+2q^{3n+1}-q^{3n-1}$ solids of $\mathcal{D}_{P}$ not contained in $\Lambda$. Moreover $|\mathcal{D}_{P}\setminus\bar{\mathcal{D}}_{P}|=(q^{2}+q+1)\left(\sum_{i=0}^{n-2}q^{6i+8}+q^{6i+5}+\ldots+q^{3i+8}+q^{3i+5}+q^{3i+2}-q^{3i+2}+2q^{3i+4}\right)+q(q+1)(2q+1)$.

Let $\mathcal{V}=\bigcup_{P\in\pi}\bar{\mathcal{D}}_{P}$ and let $\mathcal{W}=\mathcal{D}_{P}\setminus\bar{\mathcal{D}}_{P}$. First of all observe that $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ is a $q$–covering design $\mathbb{C}_{q}(3n+8,4,2)$. Indeed, from Proposition 2.1, every line of ${\rm PG}(3n+7,q)$ disjoint from $\Lambda$ is contained in exactly one element of $\mathcal{U}$. On the other hand, if $r$ is a line of ${\rm PG}(3n+7,q)$ meeting $\Lambda$ in at least a point, then $r$ is contained in $\Lambda_{P}$, for some $P\in\Pi$, and $r$ is contained in at least a solid of $\mathcal{D}_{P}$. Hence $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ is a $q$–covering design $\mathbb{C}_{q}(3n+8,4,2)$.

Let $\sigma$ be a plane of $\Pi$ and let $\Gamma$ be the hyperplane $\langle\Lambda,\sigma\rangle$ of ${\rm PG}(3n+7,q)$. Since every solid of $\mathcal{U}$ is disjoint from $\Lambda$, we have that no solid of $\mathcal{U}$ is contained in $\Gamma$. The solids of $\mathcal{V}$ not contained in $\Gamma$ are those of the set $\bigcup_{P\in\Pi,P\notin\sigma}\bar{\mathcal{D}}_{P}$. Furthermore the solids of $\mathcal{W}$ are contained in $\Gamma$. Hence there are

[TABLE]

solids of $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ not contained in $\Gamma$. Finally note that the solids of $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ contained in $\Gamma$ are those of $\bigcup_{P\in\sigma}\bar{\mathcal{D}}_{P}\cup\mathcal{W}$. Hence there are

[TABLE]

solids of $\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}$ contained in $\Gamma$. ∎

Taking into account Theorem 4.6 and [10, Theorem 1], the following result holds true.

Corollary 4.7**.**

If $n\geq 0$, then

[TABLE]

5 On $\mathcal{C}_{q}(2n,4,3)$

The main goal of this section is to give an upper bound on $\mathcal{C}_{q}(2n,4,3)$, $n\geq 4$. We begin by providing a construction in the case $n=4$.

Construction 5.1**.**

Let $\mathcal{A}$ be a $(4\times 4,2)_{q}$ MRD–code and let $\mathcal{X}=\{L(A)\;|\;A\in\mathcal{A}\}$ be the set of $q^{12}$ solids of ${\rm PG}(7,q)$ obtained by lifting the matrices of $\mathcal{A}$. Let $\Sigma^{\prime}$ be the solid of ${\rm PG}(7,q)$ containing $U_{1},U_{2},U_{3},U_{4}$. Then $\Sigma^{\prime}$ is disjoint from $\Sigma$. Let $\mathcal{P}=\{\mathcal{S}_{i}\;|\;1\leq i\leq q^{2}+q+1\}$ be a $1$–parallelism of $\Sigma$, let $\mathcal{P}^{\prime}=\{\mathcal{S}_{i}^{\prime}\;|\;1\leq i\leq q^{2}+q+1\}$ be a $1$–parallelism of $\Sigma^{\prime}$ and let $\mu:\mathcal{S}_{i}^{\prime}\in\mathcal{P}^{\prime}\longmapsto\mathcal{S}_{i}\in\mathcal{P}_{i}$ be a bijection. For a line $\ell^{\prime}$ of $\Sigma^{\prime}$, let $\Gamma_{\ell^{\prime}}$ denote the $5$–space containing $\Sigma$ and $\ell^{\prime}$. Since $\mathcal{P}^{\prime}$ is a $1$–parallelism of $\Sigma^{\prime}$, there exists a unique $j$, with $1\leq j\leq q^{2}+q+1$, such that $\ell^{\prime}\in\mathcal{S}_{j}^{\prime}$. Note that $\mu(\mathcal{S}_{j}^{\prime})=\mathcal{S}_{j}$ is a line–spread of $\Sigma$. Let $\ell$ be a line of $\mathcal{S}_{j}$ and let $\mathcal{Y}_{\ell}$ be the set of $q^{4}$ solids of $\Gamma_{\ell^{\prime}}$ (distinct from $\Sigma$) meeting $\Sigma$ exactly in $\ell$. Let $\mathcal{Z}_{\ell^{\prime}}=\bigcup_{\ell\in\mathcal{S}_{j}}\mathcal{Y}_{\ell}$. Then $\mathcal{Z}_{\ell^{\prime}}$ consists of $q^{4}(q^{2}+1)$ solids. Varying $\ell^{\prime}$ among the lines of $\Sigma^{\prime}$, we get a set

[TABLE]

consisting of $q^{4}(q^{2}+1)^{2}(q^{2}+q+1)$ solids.

Theorem 5.2**.**

The set $\mathcal{X}\cup\mathcal{Z}\cup\{\Sigma\}$ is a $q$–covering design $\mathbb{C}_{q}(8,4,3)$ of size $q^{12}+q^{4}(q^{2}+1)^{2}(q^{2}+q+1)+1$.

Proof.

Let $\pi$ be a plane of ${\rm PG}(7,q)$. If $\pi$ is disjoint from $\Sigma$, then, from Proposition 2.1, we have that $\pi$ is contained in exactly one element of $\mathcal{X}$. If $\pi$ meets $\Sigma$ in one point, say $P$, then let $\Lambda$ be the $5$–space $\langle\Sigma,\pi\rangle$ and let $\ell^{\prime}$ be the line of $\Sigma^{\prime}$ obtained by intersecting $\Sigma^{\prime}$ with $\Lambda$. Note that $\Lambda=\Gamma_{\ell^{\prime}}$. Let $\mathcal{S}_{j}^{\prime}$ be the unique line–spread of $\mathcal{P}^{\prime}$ containing $\ell^{\prime}$. Then there exists a unique line $\ell$ of $\mathcal{S}_{j}=\mu(\mathcal{S}_{j}^{\prime})$ such that $P\in\ell$ and $\pi$ is contained in $\langle\pi,\ell\rangle$, that is a solid of $\mathcal{Z}$. If $\pi$ meets $\Sigma$ in a line, say $r$, then let $\mathcal{S}_{k}$ be the unique line–spread of $\mathcal{P}$ containing $r$ and let $\Lambda$ be the $4$–space $\langle\Sigma,\pi\rangle$. Then $\Lambda\cap\Sigma^{\prime}$ is a point, which belongs to a unique line, say $r^{\prime}$, of the line–spread $\mu^{-1}(\mathcal{S}_{k})=\mathcal{S}_{k}^{\prime}$ of $\mathcal{P}^{\prime}$. Since there are $q^{2}$ solids of $\Gamma_{r^{\prime}}$ meeting $\Sigma$ exactly in $r$ and containing $\pi$, we have that in this case $\pi$ is contained in $q^{2}$ members of $\mathcal{Z}$. Finally if $\pi$ is a plane of $\Sigma$, then $\pi$ is contained in $\Sigma$. ∎

Remark 5.3**.**

Note that Theorem 5.2 generalizes the result of [10, Theorem 16].

Proposition 5.4**.**

There exists a $5$–space $\Lambda$ of ${\rm PG}(7,q)$ containing exactly $q^{4}(q^{2}+1)+1$ members of $\mathcal{X}\cup\mathcal{Z}\cup\{\Sigma\}$. Moreover every hyperplane of ${\rm PG}(7,q)$ through $\Lambda$ contains precisely $q^{4}(q^{2}+1)(q^{2}+q+1)+1$ solids of $\mathcal{X}\cup\mathcal{Z}\cup\{\Sigma\}$.

Proof.

Let $\Lambda$ be a $5$–space containing $\Sigma$. Then $\Lambda$ meets $\Sigma^{\prime}$ in a line, say $r$, and $\Lambda=\langle\Sigma,r\rangle$. The line $r$ belongs to a unique line–spread $\mathcal{S}_{i}^{\prime}$ of the $1$–parallelism $\mathcal{P}^{\prime}$ of $\Sigma^{\prime}$. Then $\mu(\mathcal{S}_{i}^{\prime})=\mathcal{S}_{i}$ is a line–spread belonging to the $1$–parallelism $\mathcal{P}$ of $\Sigma$. The $q^{4}(q^{2}+1)$ solids of $\mathcal{Z}$ lying in $\langle\Sigma,r\rangle$ meet $\Sigma$ in a line of $\mathcal{S}_{i}$ and are contained in $\Lambda$. Let $s$ be a line of $\Sigma^{\prime}$ such that $s\neq r$. In this case none of the $q^{4}(q^{2}+1)$ solids of $\mathcal{Z}$ lying in $\langle\Sigma,s\rangle$ is contained in $\Lambda$. Indeed, assume by contradiction that there is a solid $\Delta$ contained in $\Lambda\cap\langle\Sigma,s\rangle$, then $\Delta\subset\langle\Sigma,s\cap r\rangle$ and hence $\Delta\cap\Sigma$ is a plane of $\Sigma$, contradicing the fact that every solid of $\mathcal{Z}$ meets $\Sigma$ in a line. On the other hand, no solid of $\mathcal{X}$ is contained in $\Lambda$, otherwise such a solid would meet $\Sigma$ not trivially. Finally note that $\Sigma$ is a solid of $\Lambda$.

Let $\Gamma$ be a hyperplane of ${\rm PG}(7,q)$ through $\Lambda$. Then $\Gamma\cap\Sigma^{\prime}$ is a plane, say $\sigma$, containing the line $r$. Repeating the previous argument for every line of the plane $\sigma$, it turns out that there are $q^{4}(q^{2}+1)(q^{2}+q+1)$ solids of $\mathcal{Z}$ in $\Gamma$, as required. ∎

Let $\mathbb{S}_{4}$ denotes $\mathcal{X}\cup\mathcal{Z}\cup\{\Sigma\}$. Similarly to [10, Theorem 17], $\mathbb{S}_{4}$ can be used as a base for a recursive construction of a $q$–covering designs $\mathbb{C}_{q}(2n,4,3)$, $n\geq 5$.

Theorem 5.5**.**

Let $\mathbb{S}_{n}$ be a $q$–covering design $\mathbb{C}_{q}(2n,4,3)$, $n\geq 4$, such that there is a $(2n-3)$–space of ${\rm PG}(2n-1,q)$, say $\Lambda_{n}$, containing precisely $\alpha_{n}$ elements of $\mathbb{S}_{n}$ and every hyperplane of ${\rm PG}(2n-1,q)$ through $\Lambda_{n}$ contains $\beta_{n}$ members of $\mathbb{S}_{n}$. Then there exists a $q$–covering design $\mathbb{C}_{q}(2n+2,4,3)$, say $\mathbb{S}_{n+1}$, where

[TABLE]

Moreover there exists a $(2n-1)$–space of ${\rm PG}(2n+1,q)$, say $\Lambda_{n+1}$, containing $\alpha_{n+1}=|\mathbb{S}_{n}|$ elements of $\mathbb{S}_{n+1}$ and such that every hyperplane of ${\rm PG}(2n+1,q)$ through $\Lambda_{n+1}$ contains $\beta_{n+1}$ members of $\mathbb{S}_{n+1}$, where

[TABLE]

Proof.

Let $\Lambda_{n}$ be the $(2n-3)$–space of ${\rm PG}(2n+1,q)$ generated by $U_{5},\dots,U_{2n+2}$, let $\mathcal{A}$ be a $(4\times(2n-2),2)$ MRD–code and let $\mathcal{U}$ be the set of $q^{6(n-1)}$ solids obtained by lifting the matrices of $\mathcal{A}$. Let $\Gamma$ be a solid disjoint from $\Lambda_{n}$. For a line $\ell$ of $\Gamma$ there exists a $q$–covering design $\mathbb{C}_{q}(2n,4,3)$, say $\mathbb{S}_{n}$, of $\langle\Lambda_{n},\ell\rangle$ such that $\alpha_{n}$ elements of $\mathbb{S}_{n}$ are contained in $\Lambda_{n}$ and every $2(n-1)$–space of $\langle\Lambda_{n},\ell\rangle$ through $\Lambda_{n}$ contains $\beta_{n}$ members of $\mathbb{S}_{n}$. Let $\mathcal{V}_{\ell}$ be the set of solids of $\mathbb{S}_{n}$ not contained in a $2(n-1)$–space of $\langle\Lambda_{n},\ell\rangle$ through $\Lambda_{n}$. Then $\mathcal{V}_{\ell}$ consists of $|\mathbb{S}_{n}|-\beta_{n}-q(\beta_{n}-\alpha_{n})$ solids. Varying $\ell$ among the lines of $\Gamma$, we obtain the following set of solids:

[TABLE]

For a point $P$ of $\Gamma$, there are $\beta_{n}$ solids of $\mathbb{S}_{n}$ contained in $\langle\Lambda_{n},P\rangle$, among which $\alpha_{n}$ are contained in $\Lambda_{n}$. Let $\mathcal{W}_{P}$ denotes the set of $\beta_{n}-\alpha_{n}$ solids of $\mathbb{S}_{n}$ contained in $\langle\Lambda_{n},P\rangle$ and not contained in $\Lambda_{n}$. Varying $P\in\Gamma$ we get the following set of solids:

[TABLE]

Let ${\bar{\mathcal{W}}}$ be set of $\alpha_{n}$ solids of $\mathbb{S}_{n}$ contained in $\Lambda_{n}$ and let $\mathbb{S}_{n+1}=\mathcal{U}\cup\mathcal{V}\cup\mathcal{W}\cup\bar{\mathcal{W}}$. We claim that $\mathbb{S}_{n+1}$ is a $q$–covering design $\mathbb{C}_{q}(2n+2,4,3)$. Let $\pi$ be a plane of ${\rm PG}(2n+1,q)$. If $\pi$ is disjoint from $\Lambda_{n}$, then, from Proposition 2.1, there is a unique solid of $\mathcal{U}$ containing $\pi$. If $\pi$ meets $\Lambda_{n}$ in a point, then $\langle\Lambda_{n},\pi\rangle$ is a $2(n-1)$–space meeting the solid $\Gamma$ in a line, say $r$. Then there is a solid of $\mathcal{V}_{r}$ containing $\pi$. Finally, if $\pi$ shares with $\Lambda_{n}$ at least a line, then there is at least a solid of $\mathcal{W}\cup\bar{\mathcal{W}}$ containing $\pi$.

By construction it follows that

[TABLE]

In order to complete the proof, let us denote with $\Lambda_{n+1}$ the $(2n-1)$–space $\langle\Lambda_{n},s\rangle$, where $s$ is a fixed line of $\Gamma$. The solids of $\mathbb{S}_{n+1}$ that are contained in $\Lambda_{n+1}$ are those of $\mathcal{V}_{s}\cup\bigcup_{P\in s}\mathcal{W}_{P}\cup\bar{\mathcal{W}}$ that are exactly the solids of $\mathbb{S}_{n}$. Hence $\alpha_{n+1}=|\mathbb{S}_{n}|$. A hyperplane $\mathcal{H}$ of ${\rm PG}(2n+1,q)$ through $\Lambda_{n+1}$ meets $\Gamma$ in a plane, say $\sigma$, where $s\subset\sigma$. Then the solids of $\mathbb{S}_{n+1}$ contained in $\mathcal{H}$ are the solids contained in

[TABLE]

Therefore

[TABLE]

∎

Corollary 5.6**.**

[TABLE]

[TABLE]