Related papers: On large maximal partial ovoids of the parabolic q…

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…

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…

In this paper, we construct an infinite family of $\frac{q-1}{2}$-ovoids of the generalized quadrangle $Q(4,q)$, for $q\equiv 1 (\text{mod}\ 4)$ and $q>5$. Together with the examples given by Bamberg et al. and constructions provided by…

We give a computer-based proof for the non-existence of distance-$2$ ovoids in the dual split Cayley hexagon $\mathsf{H}(4)^D$. Furthermore, we give upper bounds on partial distance-$2$ ovoids of $\mathsf{H}(q)^D$ for $q \in \{2, 4\}$.

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…

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…

Two results are obtained that give upper bounds on partial spreads and partial ovoids respectively. The first result is that the size of a partial spread of the Hermitian polar space $\mathsf{H}(3, q^2)$ is at most $\left(\frac{2p^3+p}{3}…

We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3 \pmod 4$.

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…

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 provide new proofs for the non-existence of ovoids in hyperbolic spaces of rank at least four in even characteristic, and for the Hermitian polar space $\mathsf{H}(5, 4)$. We also improve the results of A. Klein on the non-existence of…

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 prove that there are no Wieferich's primes $q=2p+1$ where $p \equiv 3 \pmod{4}$ is a prime number

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…

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that…

Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest…

Given an elliptic curve $E$ over a finite field $\F_q$ of $q$ elements, we say that an odd prime $\ell \nmid q$ is an Elkies prime for $E$ if $t_E^2 - 4q$ is a quadratic residue modulo $\ell$, where $t_E = q+1 - #E(\F_q)$ and $#E(\F_q)$ is…

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…

We prove that, given a partition of the point-set of $PG(3,q), q=2^n >2$, by ovoids $\{\theta_i\}^q_{i=0}$ of $PG(3,q)$ and a line $\ell$ of $PG(3,q)$, not tangent to $\theta_0$ if $\ell^\perp$ denotes the polar of $\ell$ relative to the…

Let $H$ be a non-semisimple Hopf algebra with antipode $S$ of dimension $pq$ over an algebraically closed field of characteristic 0 where $p \le q$ are odd primes. We prove that $\Tr(S^{2p})=p^2d$ where $d \equiv pq \pmod{4}$. As a…