On \(k\)-Arcs Covering a Line in Finite Projective Planes

In a finite projective plane, a \(k\)-arc \(\mathcal{K}\) covers a line \(l_0\) if every point on \(l_0\) lies on a secant of \(\mathcal{K}\). Such \(k\)-arcs arise from determining sets of elements for which no linear \((n, q, t)\)-perfect hash families exist [1], as well as from finding sets of points in \(\mathrm{AG}(2, q)\) which determine all directions [2]. This paper provides a lower bound on \(k\) and establishes exactly when the lower bound is attained. This paper also gives constructions of such \(k\)-arcs with \(k\) close to the lower bound.