On minimal blocking sets of the generalized quadrangle Q(4,q)Q(4, q)

The generalized quadrangle $Q(4,q)$ arising from the parabolic quadric in $PG(4,q)$ always has an ovoid. It is not known whether a minimal blocking set of size smaller than $q^2 + q$ (which is not an ovoid) exists in $Q(4,q)$, $q$ odd. We present results on smallest blocking sets in $Q(4,q)$, $q$ odd, obtained by a computer search. For $q = 5,7,9,11$ we found minimal blocking sets of size $q^2 + q - 2$ and we discuss their structure. By an exhaustive search we excluded the existence of a minimal blocking set of size $q^2 + 3$ in $Q(4,7)$

Blocking sets of the Hermitian unital

It is known that the classical unital arising from the Hermitian curve in PG(2,9) does not have a 2-coloring without monochromatic lines. Here we show that for q≥4 the Hermitian curve in PG(2,q2) does possess 2-colorings without monochromatic lines. We present general constructions and also prove a lower bound on the size of blocking sets in the classical unital

Double blocking sets of size 3q-1 in PG(2,q)

The main purpose of this paper is to find double blocking sets in PG(2,q) of size less than 3q, in particular when q is prime. To this end, we study double blocking sets in PG(2,q) of size 3q-1 admitting at least two (q-1)-secants. We derive some structural properties of these and show that they cannot have three (q-1)-secants. This yields that one cannot remove six points from a triangle, a double blocking set of size 3q, and add five new points so that the resulting set is also a double blocking set. Furthermore, we give constructions of minimal double blocking sets of size 3q-1 in PG(2,q) for q=13, 16, 19, 25, 27, 31, 37 and 43. If q>13 is a prime, these are the first examples of double blocking sets of size less than 3q. These results resolve two conjectures of Raymond Hill from 1984

2-semiarcs in PG(2, q), q <= 13

A 2-semiarc is a pointset S-2 with the property that the number of tangent lines to S-2 at each of its points is two. Using some theoretical results and computer aided search, the complete classification of 2-semiarcs in PG(2, q) is given for q <= 7, the spectrum of their sizes is determined for q <= 9, and some results about the existence are proven for q = 11 and q = 13. For several sizes of 2-semiarcs in PG(2, q), q <= 7, classification results have been obtained by theoretical proofs

Dominating sets in projective planes

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order $q>81$ is smaller than $2q+2[\sqrt{q}]+2$ (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most $2q+\sqrt{q}+1$. In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3q, then it consists of the union of a blocking set and a covering set apart from a few points and lines.Comment: 19 page

Higgledy-piggledy sets in projective spaces of small dimension

This work focuses on higgledy-piggledy sets of $k$-subspaces in $\text{PG}(N,q)$, i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these $k$-subspaces with any $(N-k)$-subspace $\kappa$ of $\text{PG}(N,q)$ spans $\kappa$ itself. We highlight three methods to construct small higgledy-piggledy sets of $k$-subspaces and discuss, for $k\in\{1,N-2\}$, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small non-trivial higgledy-piggledy sets in $\text{PG}(N,q)$, $N\leqslant5$. Our main result is the existence of six lines of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in $\text{PG}(5,q)$ consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding- and graph-theoretical context.Comment: [v1] 21 pages, 1 figure [v2] 21 pages, 1 figure: corrected minor details, updated bibliograph