Related papers: Relative $m$-ovoids of elliptic quadrics

In this paper we are concerned with $m$-ovoids of the symplectic polar space ${\cal W}(2n+1, q)$, $q$ even. In particular we show the existence of an elliptic quadric of ${\rm PG}(2n+1, q)$ not polarizing to ${\cal W}(2n+1, q)$ forming a…

An $m$-ovoid of a finite polar space $\mathcal{P}$ is a set $\mathcal{O}$ of points such that every maximal subspace of $\mathcal{P}$ contains exactly $m$ points of $\mathcal{O}$. In the case when $\mathcal{P}$ is an elliptic quadric…

We construct a family of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$, the parabolic quadric of $\textup{PG}(4,q)$, for $q\equiv 3\pmod 4$. The existence of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$ was only known for $q=3, 7,$ or $11$. Our construction…

An infinite family of $(q^2+q+1)$-ovoids of $\mathcal{Q}^+(7,q)$, $q\equiv 1\pmod{3}$, admitting the group $\mathrm{PGL}(3,q)$, is constructed. The main tool is the general theory of generalized hexagons.

We generalise the work of Segre (1965), Cameron - Goethals - Seidel (1978), and Vanhove (2011) by showing that nontrivial $m$-ovoids of the dual polar spaces $DQ(2d, q)$, $DW(2d-1,q)$ and $DH(2d-1,q^2)$ ($d\ge 3$) are hemisystems. We also…

In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection…

We present a description of maximal partial ovoids of size $q^2-1$ of the parabolic quadric $\q(4,q)$ as sharply transitive subsets of $\SL(2,q)$ and show their connection with spread sets. This representation leads to an elegant explicit…

A {\em maximal partial ovoid} of a generalized quadrangle is a maximal set of points no two of which are collinear. The problem of determining the smallest size of a maximal partial ovoid in quadrangles has been extensively studied in the…

Ovoids of the non-degenerate quadric Q(4,q) of PG(4,q) have been studied since the end of the '80s. They are rare objects and, beside the classical example given by an elliptic quadric, only three classes are known for q odd, one class for…

We constuct a family of hemisystems of the parabolic quadric $\mathcal{Q}(2d, q)$, for all ranks $d \ge 2$ and all odd prime powers $q$, that admit $\Omega_3(q) \cong \mathrm{PSL}_2(q)$. This yields the first known construction for $d \ge…

An $m$-cover of the Hermitian surface $H(3,q^2)$ of $PG(3,q^2)$ is a set $\mathcal{S}$ of lines of $H(3,q^2)$ such that every point of $H(3,q^2)$ lies on exactly $m$ lines of $\mathcal{S}$, and $0<m<q+1$. Segre (1965) proved that if $q$ is…

Ovoids in $\PG(3, q)$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families are known. The first is the elliptic quadratics and the second is the Tits ovoids. In this…

We construct an infinite family of hyperovals on the Klein quadric $Q^+(5,q)$, $q$ even. The construction makes use of ovoids of the symplectic generalized quadrangle $W(q)$ that is associated with an elliptic quadric which arises as solid…

Ovoids of the Klein quadric $Q^+(5,q)$ of $\mathrm{PG}(5,q)$ have been studied in the last 40 year, also because of their connection with spreads of $\mathrm{PG}(3,q)$ and hence translation planes. Beside the classical example given by a…

Ovoids in $\PG(3, \gf(q))$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families of ovoids are known. The first is the elliptic quadratics and the second is the Tits…

Several classes of near-MDS codes of ${\rm PG}(3,q)$ are described. They are obtained either by considering the intersection of an elliptic quadric ovoid and a Suzuki-Tits ovoid of a symplectic polar space ${\cal W}(3, q)$ or starting from…

An $n\times n$ real matrix $Q$ is quasi-orthogonal if $Q^{\top}Q=qI_{n}$ for some positive real number $q$. If $M$ is a principal sub-matrix of a quasi-orthogonal matrix $Q$, we say that $Q$ is a quasi-orthogonal extension of $M$. In a…

In this paper we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space $\mathrm{H}(3,q^2)$, unifying all known infinite families. We use these conditions to provide new proofs of the…

We consider the following generalization of the decomposition theorem for polycycles. A {\em $(R,q)$-polycycle} is, roughly, a plane graph, whose faces, besides some disjoint {\em holes}, are $i$-gons, $i \in R$, and whose vertices, outside…

Ovoids of the hyperbolic quadric $Q^+(7,q)$ of $\mathrm{PG}(7,q)$ have been extensively studied over the past 40 years, partly due to their connections with other combinatorial objects. It is well known that the points of an ovoid of…