On the spectrum of minimal blocking sets in \(\text{PG}(2,q)\)
A blocking set \(B\) in PG\((2,q)\) is a set of points intersecting every line in at least one point. A blocking set \(B\) is called trivial when it contains a line and it is called minimal when none of its proper subsets still is a blocking set. This article gives a survey on the spectrum of minimal non-trivial blocking sets in PG\((2,q)\). This is the problem of determining the integers \(k\) for which minimal blocking sets of size \(k\) in PG\((2,q)\) exist. The authors have collected the results according to the intervals: (1) \(k <3(q+1)/2\) corresponding to the sizes of small minimal blocking sets, (2) \(3(q+1)/2 \leq k \leq 2q-1\), (3) \(k>2q-1\), and (4) large values of \(k\). In the final subsection, minimal non-trivial blocking sets in planes of small order are discussed. Regarding the different cases, there is hope that small minimal non-trivial blocking sets can be characterized. Regarding (2), there seems to be some structure in the minimal non-trivial blocking sets of these sizes, particularly when \(k\) is only slightly larger than \(3(q+1)/2\). There are in case (3) many blocking sets of size between \(2q-1\) and \(3q-3\), but no examples of minimal non-trivial blocking sets of size \(cq\) are known, where \(c\) is a constant larger than 3. There are several constructions giving minimal blocking sets of size \(cq \log q\). Very few constructions give larger blocking sets; a new construction is presented in this article. In the second section, two general construction methods for minimal blocking sets are discussed. They can be used to construct blocking sets of cardinalities belonging to distinct intervals mentioned above.
scientific article; zbMATH DE number 1972492