Small complete caps in PG\((2,q)\), for \(q\) an odd square (Q1841865)

A cap \(K\) in a projective space PG\((d,q)\) of dimension \(d\) over the finite field GF\((q)\) is a set of points no three of which are collinear. A cap \(K\) is complete if any point outside \(K\) is a bisecant of \(K\). A cap having \(n\) points is called an \(n\)-cap. For \(d=2\) caps are also called arcs. By using a set of two conics the author constructs a class of \(4(\sqrt q - 1)\)-arcs in PG\((2,q)\) for \(q=m^2\), \(m = p^h\), \(p\) an odd prime. By using the computer completeness and uncompleteness of several of these arcs are proved. Based on this data the main theorem states that for any odd square \(q\) with \(q \leq 1681\) or \(q = 2401\) there exists a complete \(4(\sqrt q - 1)\)-arc in PG\((2,q)\).