On upper bounds on the smallest size of a saturating set in a projective plane

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 upper bound on the smallest size $s(2,q)$ of a saturating set in $\Pi _{q}$ is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln (q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any constant $c\ge 1$ a random point set of size $k$ in $\Pi _{q}$ with $2c\sqrt{(q+1)\ln(q+1)}+2\le k<\frac{q^{2}-1}{q+2}\thicksim q$ is a saturating set with probability greater than $1-1/(q+1)^{2c^{2}-2}.$ Our probabilistic approach is also applied to multiple saturating sets. A point set $S\subset \Pi_{q}$ is $(1,\mu)$-saturating if for every point $Q$ of $\Pi _{q}\setminus S$ the number of secants of $S$ through $Q$ is at least $\mu$, counted with multiplicity. The multiplicity of a secant $\ell$ is computed as ${\binom{{\#(\ell \,\cap S)}}{{2}}}.$ The following upper bound on the smallest size $s_{\mu }(2,q)$ of a $(1,\mu)$-saturating set in $\Pi_{q}$ is proved: \begin{equation*} s_{\mu }(2,q)\leq 2(\mu +1)\sqrt{(q+1)\ln (q+1)}+2\thicksim 2(\mu +1)\sqrt{ q\ln q}\,\text{ for }\,2\leq \mu \leq \sqrt{q}. \end{equation*} By using inductive constructions, upper bounds on the smallest size of a saturating set (as well as on a $(1,\mu)$-saturating set) in the projective space $PG(N,q)$ are obtained. All the results are also stated in terms of linear covering codes.Comment: 15 pages, 24 references, misprints are corrected, Sections 3-5 and some references are adde

On sizes of complete arcs in PG(2,q)

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, 461,463,467,479,487,491,499,529,563,569,571,577,587,593,599,601,607,613,617,619,631, 641,661,673,677,683,691, 709}, T2={4597,4703,4723,4733,4789,4799,4813,4831,5003,5347,5641,5843,6011,8192}. From these new bounds it follows that for q <= 2593 and q=2693,2753, the relation t_{2}(2,q) < 4.5\sqrt{q} holds. Also, for q <= 4561 we have t_{2}(2,q) < 4.75\sqrt{q}. It is showed that for 23 <= q <= 4561 and q\in T2\cup {2^{14},2^{15},2^{18}}, the inequality t_{2}(2,q) < \sqrt{q}ln^{0.75}q is true. Moreover, the results obtained allow us to conjecture that this estimate holds for all q >= 23. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms. Also new constructions of complete arcs are proposed. These constructions form families of k-arcs in PG(2,q) containing arcs of all sizes k in a region k_{min} <= k <= k_{max} where k_{min} is of order q/3 or q/4 while k_{max} has order q/2. The completeness of the arcs obtained by the new constructions is proved for q <= 1367 and 2003 <= q <= 2063. There is reason to suppose that the arcs are complete for all q > 1367. New sizes of complete arcs in PG(2,q) are presented for 169 <= q <= 349 and q=1013,2003.Comment: 27 pages, 4 figures, 5 table

The use of blocking sets in Galois geometries and in related research areas

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems