New hemisystems of the Hermitian surface

TL;DR

This paper introduces new hemisystems of the Hermitian surface in projective space, constructed for primes of a specific form, and demonstrates their applications in coding theory and graph theory.

Contribution

It provides novel examples of hemisystems in $PG(3,p^2)$ for primes of the form $p=1+4a^2$, expanding the known constructions using maximal curves.

Findings

New hemisystems for primes p=1+4a^2

Construction of two-weight linear codes from hemisystems

Generation of strongly regular graphs from hemisystems

Abstract

Finding Hemisystems is a challenging problem and just few examples arising from the Hermitian surface are known. A recent method to obtain Hemisystems is based on using maximal curves. Along this side of research, we provide new examples of Hemisystems in $PG(3,p^2)$, for each prime of the form $p=1+4a^2$, with $ a $ integer. Last, we use these results to obtain two weight linear codes and strongly regular graphs.