Minimal blocking sets in PG\((2,q)\) arising from a generalized construction of Megyesi (Q1022876)

A well known construction of minimal blocking sets is due to Megyesi: take \(d\) to be a divisor of \(q-1\) and assume \(G\) to be a multiplicative subgroup of \(GF(q)^*\) with size \(d\). Then, the set \[ U=\{(0,0)\}\cup\{(0,h): h\not\in G\}\cup\{(g,0):g\in G\} \] determines a set \(D\) of \(q+1-d\) directions in \(AG(2,q)\). The set \(B=U\cup D\) turns out to be a minimal blocking set of size \(2q+1-d\) of \(PG(2,q)\). When \(d\) divides \(q\), an analogous construction might be performed using two parallel lines in \(AG(2,q)\) and an additive subgroup of \(GF(q)\). Observe that in the first case \(B\) is contained in a triangle [indeed, when \(d=(q-1)/2,\) the construction yields the so called \textit{projective triangle}], whereas in the second case \(B\) is included into a star (and, for \(d=q/2\) we speak of a \textit{projective triad}). In this paper the authors generalize the aforementioned construction by considering set of points placed on \(n\) lines of the affine plane. The first result is for \(n=3\). {\textbf{Construction 2.1}}: Let \(s\geq 3\) be a divisor of \(q-1\) and consider a multiplicative subgroup \(G\) of \(GF(q)^*\) with index \(s\). Take \(\alpha\in GF(q)^*\) as an element such that \(G,\alpha G,\alpha^2 G,\dots,\alpha^{s-1}G\) are all the cosets of \(G\) and introduce three non-empty subsets \(I,J,K\subseteq{\mathbb{Z}_s}\) with \(|I|+|J|+|K|=s\). Define \[ U=\{(0,x):x\in\alpha^i G,i\in I\}\cup\{(x,0):x\in\alpha^j G,j\in J\}\cup\{(x,x):x\in\alpha^k G,k\in K\}\cup\{(0,0)\} \] and let \(D\) be the set of directions determined by \(U\). If \(|D|<q+1\), then \(B=U\cup D\) is a minimal blocking set. This result is then used to obtain some examples of minimal blocking sets of \(PG(2,q)\) with prescribed size. A similar construction holds also for \(n\geq 4\). A blocking set contained in the union of \(n\) concurrent affine lines and the line at infinity will be obtained. {\textbf{Construction 3.1}} Consider a multiplicative subgroup \(G\) of \(GF(q)^*\) with index \(s\geq n\) and take \(\alpha\in GF(q)^*\) such that \(\alpha^i G\) with \(i=0,\ldots,s-1\) are the cosets of \(G\). Let \(m_1=\infty\) and \(\{m_2,\ldots,m_n\}\subseteq GF(q)\) be the set of the slopes of the remaining \(n-1\) lines. Form \(n\) non empty subsets \(A_1,A_2,\ldots, A_n\) in \({\mathbb{Z}_s}\) such that \(|A_1|+|A_2|+\cdots+|A_n|=s\). Then, \[ U=\{(0,0)\}\cup\{(0,x):x\in\alpha^a: a\in A_1\}\cup\bigcup_{i=2}^n\{ (x,m_ix): x\in\alpha^a G, a\in A_i\} \] determines a set of directions \(D\). If \(|D|<q+1\), then \(B=U\cup D\) is a minimal blocking set. The aforementioned constructions, with \(s\) chosen as relatively small with respect to \(q\), provide minimal blocking sets with size \[ |B|\geq\left(2-{2\over 9}\right)q+O(\sqrt{q}) \] and also \[ |B|\geq\left(2-{{(n-1)!}\over{n^{n-1}}}\right)q+O(\sqrt{q}). \] In the last section of the paper, constructions for deriving blocking sets in \(PG(2,q^h)\) from existing blocking sets in \(PG(2,q)\) are investigated and applied to the previously obtained sets; these are used to show that if there is a minimal blocking set of size \(2q-x\) in \(PG(2,q)\), then there are also minimal blocking sets of size \(2q^h-x\) and \(2q^h-x+1\) in \(PG(2,q^h)\) which are not necessarily of Rédei type.