Related papers: Ovoidal fibrations in $PG(3,q), q$ even

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…

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…

The Desarguesian ovoids in the orthogonal polar space $Q^+(7,q)$ with $q$ even have first been introduced by Kantor by examining the $8$-dimensional absolutely irreducible modular representations of $\text{PGL}(2,q^3)$. We investigate this…

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 consider the cyclic presentation of $PG(3,q)$ whose points are in the finite field $\mathbb{F}_{q^4}$ and describe the known ovoids therein. We revisit the set $\mathcal{O}$, consisting of $(q^2+1)$-th roots of unity in…

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…

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…

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…

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…

In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces ${\cal W}(3, q)$, $q$ odd square, $q \not\equiv 0 \pmod{3}$, ${\cal W}(5, q)$ and of the Hermitian polar spaces…

Let $K=Q(\sqrt{-\ell})$ be an imaginary quadratic field with ring of integers $\O_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$ and $C=[n, k]$ be a linear code defined over $\O_K/2\O_K$. The level $\ell$ theta…

Let $\ell$ be an odd prime and $K$ a field of characteristic different from $\ell$. Let $\bar{K}$ be an algebraic closure of $K$. Assume that $K$ contains a primitive $\ell$th root of unity. Let $n \ne \ell$ be another odd prime. Let $f(x)$…

For $\ell = 3$ and 5 it is known that every odd, irreducible, 2-dimensional representation of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ and determinant equal to the cyclotomic character must "come from" the $\ell$-torsion points of an…

The geometry of the real four-qubit Pauli group, being embodied in the structure of the symplectic polar space W(7,2), is analyzed in terms of ovoids of a hyperbolic quadric of PG(7,2), the seven-dimensional projective space of order two.…

An ovoid of a dual polar space is a point set meeting every line in exactly one point. For the symplectic dual polar space DW(6,q), Cooperstein and Pasini have recently proved no ovoid exists if q is odd. Earlier, Shult has proved the same…

We consider the orbits of the group $G=PGL_2(q)$ on the points, lines and planes of the projective space $PG(3,q)$ over a finite field $\mathbb F_q$ of characteristic different from $2$ and $3$. The points of $PG(3,q)$ can be identified…

We use the representation $T_2(O)$ for $\q(4,q)$ to show that maximal partial ovoids of $\q(4,q)$ of size $q^2-1$, $q=p^h$, $p$ odd prime, $h > 1$, do not exist. Although this was known before, we give a slightly alternative proof, also…

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…

Let ${\cal Q}^-(2n+1,q)$ be an elliptic quadric of ${\rm PG}(2n+1,q)$. A relative $m$-ovoid of ${\cal Q}^-(2n+1,q)$ (with respect to a parablic section ${\cal Q} := {\cal Q}(2n,q) \subset {\cal Q}^-(2n+1,q)$) is a subset $\cal R$ of points…

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give…