On intersection sets in Desarguesian affine spaces (Q1568781)

This paper concentrates on finding lower bounds on the size of \(t\)-fold blocking sets with respect to hyperplanes or \(t\)-intersection sets in \(AG(n, q)\), the n-dimensional Desarguesian affine space over the finite field \(GF(q)\), where \(q=p^h\) for some prime \(p\). A set of points \(S\) is a \(t\)-fold blocking set with respect to hyperplanes if every hyperplane contains at least \(t\) points of \(S\). Such sets are also known as \(t\)-intersection sets or intersection sets with multiplicity \(t\). The main result in the paper states that for \(t< q\), a \(t\)-fold blocking set with respect to hyperplanes in \(AG(n, q)\) has at least \((t+n-1)(q-1)+k\) points provided that there exists a \(j\) such that \(k-1\leq j < t\) and the binomial coeficient \({k-n-t \choose j}\neq 0\pmod p\). Furthermore, for \(t<q\), a \(t\)-fold blocking set with respect to hyperplanes in \(AG(n, q)\) has at least \((t+n-1)q-n+1\) points provided \({-n \choose t-1}\neq 0\pmod p\).