Related papers: Algebraic constructions of complete $m$-arcs

We show how the theory of affine geometries over the ring ${\mathbb Z}/\langle q - 1\rangle$ can be used to understand the properties of toric and generalized toric codes over ${\mathbb F}_q$. The minimum distance of these codes is strongly…

We show that the set of F_q-rational points of either certain Fermat curves or certain F_q-Frobenius non-classical plane curves is a complete (k,d)-arc in P^2(F_q), where k and d are respectively the number of F_q-rational points and the…

In recent years, many connections have been made between minimal codes, a classical object in coding theory, and other remarkable structures in finite geometry and combinatorics. One of the main problems related to minimal codes is to give…

To find the non-standard binary Golay complementary sequences (GCSs) of length $2^{m}$ or theoretically prove the nonexistence of them are still open. Since it has been shown that all the standard $q$-ary (where $q$ is even) GCSs of length…

Let $\A$ be the incidence matrix of lines and points of the classical projective plane $PG(2,q)$ with $q$ odd. With respect to a conic in $PG(2,q)$, the matrix $\A$ is partitioned into 9 submatrices. The rank of each of these submatices…

The cage problem asks for the smallest number $c(k,g)$ of vertices in a $k$-regular graph of girth $g$ and graphs meeting this bound are known as cages. While cages are known to exist for all integers $k \ge 2$ and $g \ge 3$, the exact…

We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$…

This research focuses on constructing $q$-ary functions for complete complementary codes (CCCs) with flexible parameters. Most existing work has primarily identified sufficient conditions for $q$-ary functions related to $q$-ary CCCs. To…

In this paper we present a complete characterization of the smallest sets which block all the simple perfect matchings in a complete convex geometric graph on $2m$ vertices. In particular, we show that all these sets are caterpillar graphs…

We study those $(2,m,n)$-groups which are almost simple and for which the absolute value of the Euler characteristic is a product of two prime powers. All such groups which are not isomorphic to $PSL_2(q)$ or $PGL_2(q)$ are completely…

Let $n$ be a positive integer. Denote by $\mathrm{PG}(n,q)$ the $n$-dimensional projective space over the finite field $\mathbb{F}_q$ of order $q$. A blocking set in $\mathrm{PG}(n,q)$ is a set of points that has non-empty intersection with…

An arc is a subset of $\mathbb F_q^2$ which does not contain any collinear triples. Let $A(q,k)$ denote the number of arcs in $\mathbb F_q^2$ with cardinality $k$. This paper is primarily concerned with estimating the size of $A(q,k)$ when…

A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth…

We consider the following feasibility problem: given an integer $n \geq 1$ and an integer $m$ such that $0 \leq m \leq \binom{n}{2}$, does there exist a line graph $L = L(G)$ with exactly $n$ vertices and $m$ edges ? We say that a pair…

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the…

We show that the perfect Euler brick (perfect cuboid) problem is equivalent to the following elementary question: do there exist coprime integers $a, b, m, n$ such that the two expressions $(2(a^2-b^2)mn)^2 + ((a^2+b^2)(m^2-n^2))^2$ and…

We consider point sets in the $m$-dimensional affine space $\mathbb{F}_q^m$ where each squared Euclidean distance of two points is a square in $\mathbb{F}_q$. It turns out that the situation in $\mathbb{F}_q^m$ is rather similar to the one…

A connected planar cubic graph is called an $m$-barrel fullerene and denoted by $F(m,k)$, if it has the following structure: The first circle is an $m$-gon. Then $m$-gon is bounded by $m$ pentagons. After that we have additional k layers of…

In 1974, J. Thas constructed a new class of maximal arcs for the Desarguesian plane of order $q^2$. The construction relied upon the existence of a regular spread of tangent lines to an ovoid in $\PG(3,q)$ and, in particular, it does apply…

We are interested in the quantity $\rho$(q, g) defined as the smallest positive integer such that r $\ge$ $\rho$(q, g) implies that any absolutely irreducible smooth projective algebraic curve defined over F q of genus g has a closed point…