Cyclic arcs of Singer type and strongly regular Cayley graphs over finite fields

In \cite{M18}, the first author gave a construction of strongly regular Cayley graphs on the additive group of finite fields by using three-valued Gauss periods. In particular, together with the result in \cite{BLMX}, it was shown that there exists a strongly regular Cayley graph with negative Latin square type parameters $(q^6,r(q^3+1),-q^3+r^2+3r,r^2+r)$, where $r=M(q^2-1)/2$, in the following cases: (i) $M=1$ and $q\equiv 3\,(\mod{4})$; (ii) $M=3$ and $q\equiv 7\,(\mod{24})$; and (iii) $M=7$ and $q\equiv 11,51\,(\mod{56})$. The existence of strongly regular Cayley graphs with the above parameters for odd $M>7$ was left open. In this paper, we prove that if there is an $h$, $1\le h\le M-1$, such that $M\,|\,(h^2+h+1)$ and the order of $2$ in $({\bf Z}/M{\bf Z})^\times$ is odd,then there exist infinitely many primes $q$ such that strongly regular Cayley graphs with the aforementioned parameters exist.