The values \(\sqrt {2q}\) and \(\log _{2}q\):\, their relationship with \(k\)-arcs. (Q2715986)

Let \(K = \{P_1,\dots ,P_k\}\) be a \(k\)-arc in a projective plane of order \(q\). Denote by \(a_i\) the number of points that are covered by secants of \(\{P_1,\dots ,P_i\}\), but not by secants of \(\{P_1,\dots ,P_{i-1}\}\). Then \(b_i \leq a_i \leq c_i\), where \(b_i = qi - q - i^3/2 + 3i^2 - 13i/2 + 5\) and \(c_i = qi - q - i^2 +3i -1\). These estimates are used to show that if \(k = [\sqrt {2q}] + 2\), then secants of \(K\) fill about half the plane. One also gets an estimate of \(k\) for the case when \(K\) is complete. In the case \(k=[\sqrt {2q}] + 2\) the author estimates the number of \(k\)-arcs needed to cover the whole plane.