Classical arcs in PG\((r,q)\) for \(23\leq q\leq 29\)
Let \(PG(r,q)\) be the \(r\)-dimensional projective space over Galois field \(F_q\). A \(k\)-arc \(K\) in \(PG(r,q)\) is a set of \(k\) points such that no \(r+1\) points of \(K\) lie in a hyperplane. A normal rational curve in \(PG(r,q)\) \((r\leq q-2)\) is a set of \(q+1\) points which is (projectively) isomorphic to \(\{(1,t, \dots,t^r)\); \(t\in F_q\cup \{\infty\}\}\). An arc is called classical or rational if it lies on a normal rational curve. Let \(m(r,q)\) be the maximum size of arcs in \(PG(r,q)\). The authors are interested in modifying B. Serge's conjecture on arcs: Let \(1\leq r\leq q-2\). (1) If \(q\) is odd, then \(m(r,q)=q+1\) and every \(m(r,q)\)-arc in \(PG(r,q)\) is classical. (2) If \(q\) is even, then \(m(r,q)=q+1\) except for \(m(2,q)=m (q-2,q)=q+2\), and every \(m(r,q)\)-arc in \(PG (r,q)\) is classical provided \(4\leq r\leq q-5\). The main results of this article, together with results for \(11\leq q\leq 19\) [see \textit{J. M. Chao} and \textit{H. Kaneta}, Discrete Math. 174, No. 1-3, 87-94 (1997; Zbl 0892.51007)] imply the following: Serge's conjecture holds for \(11\leq q\leq 27\). The main results are computational. All programmes are written in the language C.
scientific article; zbMATH DE number 1565956