Arcs and ovals in the Hermitian and Ree unitals (Q749543)

An s-arc in a \(2-(q^ 3+1,q+1,1)\) design is a set of s points meeting no block in more than 2 points. It is called an oval if \(s=q^ 2+1\) and q is odd or if \(s=q^ 2\) and q is even. The 2-transitive examples of such designs are the Hermitian and the Ree unitals (where \(q-3^{2m+1})\). For \(q>2\), the Hermitian unitals are shown not to possess ovals (for q odd, this is ascribed to Andriamanalimanana). For \(q=3^{2m+1}\), the Ree unitals are shown to possess \(3q+1\)-arcs; they cannot be extended to ovals if \(m>0\). Some relations with codes and constructions of other 2- designs are discussed.