Related papers: Resolving sets and semi-resolving sets in finite p…
In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following…
New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853 <= q <= 4561 and q\in T1\cup T2 where T1={173,181,193,229,243,257,271,277,293,343,373,409,443,449,457,…
In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space ${\rm PG}(3,q^3)$, one can find three $\mathbb F_q$-subgeometries such that the union of their point sets is a strong blocking set. This proves the…
Let $G$ be a connected graph and $d(a,b)$ be the distance between the vertices $a$ and $b$. A subset $U =\{u_1,u_2,\cdots,u_k\}$ of the vertices is called a resolving set for $G$ if for every two distinct vertices $a,b \in V(G)$, there is a…
In this paper it has been verified, by a computer-based proof, that the smallest size of a complete arc is 14 in PG(2,31) and in PG(2,32). Some examples of such arcs are also described.
In the projective planes $\mathrm{PG}(2,q)$, we collect the smallest known sizes of complete arcs for the regions \begin{align*} &\mbox{all } q\le160001,~~ q \mbox{ prime power};\\ &Q_{4}=\{34 \mbox{ sporadic }q'\mbox{s in the interval…
We determine the partition dimension of the incidence graph $G(\Pi_q)$ of the projective plane $\Pi_q$ up to a constant factor $2$ as $(2+o(1))\log_2{q}\leq \mathrm{pd}(G(\Pi_q))\leq (4+o(1))\log_2{q}.$
In this paper we present and analyze computational results concerning small complete caps in the projective spaces $\mathrm{PG}(N,q)$ of dimension $N=3$ and $N=4$ over the finite field of order $q$. The results have been obtained using…
Let $G$ be a finite, connected graph of order of at least 2, with vertex set $V(G)$ and edge set $E (G)$. A set $S$ of vertices of the graph $G$ is a doubly resolving set for $G$ if every two distinct vertices of $G$ are doubly resolved by…
A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen…
The subject of this paper is the study of small complete arcs in $\mathrm{PG}(2,q)$, for $q$ odd, with at least $(q+1)/2$ points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal…
A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q…
We provide classification results for translation generalized quadrangles of order less or equal to $64$, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in…
An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J.…
The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each of these designs is embeddable in a unique…
In this paper we present the results from a program developed by the author that finds the unitals of the known 193 projective planes of order 25.. There are several planes for which we have not found any unital. One or more than one…
We improve on the lower bound of the maximum number of planes in $\operatorname{PG}(8,q)\cong\F_q^{9}$ pairwise intersecting in at most a point. In terms of constant dimension codes this leads to $A_q(9,4;3)\ge q^{12}+…
The convexity number of a set $X \subset \mathbb{R}^2$ is the minimum number of convex subsets required to cover it. We study the following question: what is the largest possible convexity number $f(n)$ of $\mathbb{R}^2 \setminus S$, where…
We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in…
Let $m$ be a positive integer, $q$ be a prime power, and $\mathrm{PG}(2,q)$ be the projective plane over the finite field $\mathbb F_q$. Finding complete $m$-arcs in $\mathrm{PG}(2,q)$ of size less than $q$ is a classical problem in finite…