New hemisystems of the Hermitian surface

Abstract

Constructing hemisystems of the Hermitian surface is a well known, apparently difficult, problem in Finite geometry. So far, a few infinite families and some sporadic examples have been constructed. One of the different approaches relies on the Fuhrmann-Torres maximal curve and provides a hemisystem in \(PG(3,p^2)\) for every prime p of the form \(p=1+4a^2\), a even. Here we show that this approach also works in \(PG(3,p^2)\) for every prime \(p=1+4a^2\), a odd. The resulting hemisystem gives rise to two weight linear codes and strongly regular graphs whose properties are also investigated.