New examples of complete k-arcs in PG(2,q) (Q792624)

This paper generalizes the result of \textit{V. Abatangelo} [Ars Comb. 16, 103-111 (1983)], in which complete \((q+8)/3\)-arcs were constructed in \(PG(2,2^ h)\), h at least 6 and even. This was the first infinite class of complete arcs constructed whose size is less than q/2. Here, for s but not 2s dividing q-1, with \(s\geq 3\) and \(q>[(s-1)^ 2+(s^ 4-4s^ 3+8s^ 2+1)^{1/2}]^ 2,\) a class of complete k-arcs in PG(2,q) is constructed, where \(k=u(q-1)/s+3\) and u satisfies certain arithmetic conditions. It is shown that, for q odd, \((q-1)/4+3\leq k\leq(q-1)/2+3;\) for q even, \((q-1)/3+3\leq k\leq(q-1)/2+3.\) When \(s=3\), the arcs constructed are those of Abatangelo and achieve the latter lower bound. It still seems a difficult problem to determine the correct size of the smallest possible complete arc.