Small proper double blocking sets in Galois planes of prime order (Q941328)

A double blocking set on \(\text{PG}(2,q)\) is a set \(B\) of points that intersects every line in at least two points. A double blocking set \(B\) is called proper if \(| B\cap\ell| \leq p-1\) for every line \(\ell\), i.e. the complement of \(B\) is again a double blocking set. The smallest known example of a proper blocking set in \(\text{PG}(2,p)\), \(p\) prime, was the disjoint union of two triangles of size \(3(p+1)/2\), i.e. a blocking set of size \(3(p+1)\). In this note, the authors construct proper blocking sets in \(\text{PG}(2,p)\) of size \(3p+1\) for each prime \(p\equiv3\pmod{4}\) and of size \(3p+2\) for each prime \(p\geq7\).