Related papers: On line-parallelisms of PG(3, q)

A quadratic line complex is a three-parameter family of lines in projective space P^3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P^3 and taking all lines of the complex passing through p we…

Every symplectic spread of PG(3,q), or equivalently every ovoid of Q(4,q), is shown to give rise to a certain family of permutation polynomials of GF(q) and conversely. This leads to an algebraic proof of the existence of the Tits-Luneburg…

Due to the applications in network coding, subspace codes and designs have received many attentions. Suppose that $k\mid n$ and $V(n,q)$ is an $n$-dimensional space over the finite field $\mathbb{F}_{q}$. A $k$-spread is a…

In this paper, we study KM-arcs in $PG(2, q)$, the Desarguesian projective plane of order $q$. A KM-arc A of type $t$ is a natural generalisation of a hyperoval: it is a set of $q + t$ points in $PG(2, q)$ such that every line of $PG(2,q)$…

This article looks at the Bose representation of $PG(2,q^3)$ as a 2-spread of $PG(8,q)$. It is shown that an $\mathbb F_q$-subline of $PG(2,q^3)$ corresponds to a 2-regulus, and an $\mathbb F_q$-subplane corresponds to a Segre variety…

For $q = p^n$ with $p$ an odd prime, the projective linear group $PGL(2,q)$ can be seen as the stabilizer of a conic $O$ in a projective plane $\pi = PG(2,q)$. In that setting, involutions of $PGL(2,q)$ correspond bijectively to points of…

The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed in Li and Pott (arXiv:2003.06678v1), which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in…

Let $G$ be a graph that is topologically embedded in the plane and let $\mathcal{A}$ be an arrangement of pseudolines intersecting the drawing of $G$. An aligned drawing of $G$ and $\mathcal{A}$ is a planar polyline drawing $\Gamma$ of $G$…

A partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in…

In $\PG(2,q^3)$, let $\pi$ be a subplane of order $q$ that is exterior to $\li$. The exterior splash of $\pi$ is defined to be the set of $q^2+q+1$ points on $\li$ that lie on a line of $\pi$. This article investigates properties of an…

A $3$-Prismatoid is the convex hull of two convex polygons $A$ and $B$ which lie in parallel planes $H_A, H_B\subset\mathbb{R}^3$. Let $A'$ be the orthogonal projection of $A$ onto $H_B$. A prismatoid is called nested if $A'$ is properly…

In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces $\operatorname{PG}(n,q)$ such that every plane is incident with at least one line of the set. This is the first main open problem…

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)$ and $D^{Q}(G)$, respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of…

A $4$-general set in ${\rm PG}(n,q)$ is a set of points of ${\rm PG}(n,q)$ spanning the whole ${\rm PG}(n,q)$ and such that no four of them are on a plane. Such a pointset is said to be complete if it is not contained in a larger…

We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent…

In this paper, we show that a small minimal k-blocking set in PG(n, q3), q = p^h, h >= 1, p prime, p >=7, intersecting every (n-k)-space in 1 (mod q) points, is linear. As a corollary, this result shows that all small minimal k-blocking…

Betten and Riesinger constructed Parallelisms of $\mathop{\rm PG}(3,\mathbb R)$ with automorphism group $\mathop{\rm SO}(3,\mathbb R)$ by applying the reducible $\mathop{\rm SO}(3,\mathbb R)$-action to a rotational Betten spread. This was…

We show that one-dimensional superspace is isomorphic to a non-trivial but consistent limit as $q\to-1$ of the braided line. Supersymmetry is identified as translational invariance along this line. The supertranslation generator and…

We examine various perspectives on the decay of correlation for the uniform distribution over proper $q$-edge colorings of graphs with maximum degree $\Delta$. First, we establish the coupling independence property when $q\ge 3\Delta$ for…

We give a complete classification, up to isometric isomorphism and scaling, of $4$-dimensional metric Lie algebras $(\mathfrak{g},\langle \cdot,\cdot \rangle)$ that admit a non-zero parallel skew-symmetric endomorphism. In particular, we…