New hemisystems of the Hermitian surface

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 , p2) for every prime p of the form p= 1 + 4 a2, a even. Here we show that this approach also works in PG(3 , p2) for every prime p= 1 + 4 a2, a odd. The resulting hemisystem gives rise to two weight linear codes and strongly regular graphs whose properties are also investigated.