Search

In a projective plane over a finite field, complete $(k,n)$-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with $k$ close to the order of the plane, is considered a hard problem. A natural candidate to be a small $(k,n)$-arc with few characters is the set $\Omega(\mathcal{C})$ of the points of a plane curve $\mathcal{C}$ of degree $n$ (containing no linear components) such that some line meets $\mathcal{C}$ transversally in the plane, i.e. in $n$ pairwise distinct points. Let $\mathcal{C}$ be either the Hermitian curve of degree $q+1$ in $\mathrm{PG}(2,q^{2r})$ with $r\ge 1$, or the rational BKS curve of degree $q+1$ in $\mathrm{PG}(2,q^r)$ with $q$ odd and $r\ge 1$. Then $\Omega(\mathcal{C})$ has four and seven characters, respectively. Furthermore, $\Omega(\mathcal{C})$ is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that $\Omega(\mathcal{C})$ is complete for $r\ge 4$. For the rational BKS curve, $\Omega(\mathcal{C})$ is complete if and only if $r$ is even. If $r$ is odd then the uncovered points by the $(q+1)$-secants to $\Omega(\mathcal{C})$ are exactly the points in $\mathrm{PG}(2,q)$ not lying in $\Omega(\mathcal{C})$. Adding those points to $\Omega(\mathcal{C})$ produces a complete $(k,q+1)$-arc in $\mathrm{PG}(2,q^r)$, with $k=q^r+q$. The above results do not hold true for $r=2$ and there remain open the case $r=3$ for the Hermitian curve, and the cases $r=3,4$ for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for $\mathrm{PGL}(2,q)$.

The extended coset leader weight enumerator of the generalized Reed-Solomon $[q + 1, q - 3, 5]_q$ code is computed. The computation is considered as a question in finite geometry. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in three space determines a rational function of degree at most three and vice versa. Furthermore the double point scheme of a rational function is studied. The pencil of a true passant of the twisted cubic, not in an osculation plane gives a curve of genus one as double point scheme. With the Hasse-Weil bound on Fq-rational points we show that there is a 3-plane containing the passant.

In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph $H$, the maximum number $k$ for which there is a balanced rainbow-free $k$-coloring of $H$ is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane $\mathrm{PG}(2,q)$ for all $q$. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.

We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\leq \frac38p+1$, a small $t$-fold (weighted) $(n-k)$-blocking set of $\mathrm{PG}(n,p)$, $p$ prime, must contain the weighted sum of $t$ not necessarily distinct $(n-k)$-spaces., Comment: 21 pages

In this paper, we prove a stability result on k mod p multisets of points in PG(2,q), q = p^h. The particular case k=0 is used to describe small weight codewords of the code generated by the lines of PG(2, q), as linear combination of few lines. Earlier results proved this for codewords with weight less than 2.5q, while our result is valid until cq sqrt(q). It is sharp when 27=4. When q is a prime, De Boeck and Vandendriessche constructed a codeword of weight 3p-3 that is not the linear combination of three lines. We characterise their example.

In the $m$-dimensional affine space $AG(m,q)$ over the finite field $\mathbb{F}_q$ of odd order $q$, the analogous of the Euclidean distance gives rise to a graph $\mathfrak{G}_{m,q}$ where vertices are the points of $AG(m,q)$ and two vertices are adjacent if their (formal) squared Euclidean distance is a square in $\mathbb{F}_q$ (including the zero). In 2009, Kurz and Meyer made the conjecture that if $m$ is even then $\mathfrak{G}_{m,q}$ is a strongly regular graph. In this paper we prove their conjecture., Comment: 14 pages

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of ${\rm PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre-Korchm\'aros on Desargues configurations with perspective triangles inscribed in a conic.

A permutation of the point set of the affine space $AG(n,q)$ is called an integral automorphism if it preserves the integral distance defined among the points. In this paper, we complete the classification of the integral automorphisms of $AG(n,q)$ for $n\ge 3$.