In [32], 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 [4], 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 $q\equiv 3(\mathrm{mod}4)$; (ii) $M=3$ and $q\equiv 7(\mathrm{mod}24)$; and (iii) $M=7$ and $q\equiv 11,51(\mathrm{mod}56)$. 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, $1⩽h⩽M-1$, such that $M|(h^2+h+1)$ and the order of 2 in ${(\mathbb{Z}/M\mathbb{Z})}^{\times }$ is odd, then there exist infinitely many primes q such that strongly regular Cayley graphs with the aforementioned parameters exist.

在[32]中，第一作者通过使用三值高斯周期在有限域的可加群上构造了强正则凯莱图。特别地，连同[4]中的结果，表明存在具有负拉丁方型参数的强正则凯莱图$(q^6,r(q^3+1),-q^3+r^2+3r,r^2+r)$， 在哪里 $r=米(q^2-1)/2$, 在以下情况下： (i) $米=1$ 和 $q\equiv 3(\mathrm{模组}4)$; (二)$米=3$ 和 $q\equiv 7(\mathrm{模组}24)$; (iii)$米=7$ 和 $q\equiv 11,51(\mathrm{模组}56)$. 具有上述奇数参数的强正则凯莱图的存在性$米>7$被打开了。在本文中，我们证明如果存在一个h，$1⩽H⩽米-1$，使得 $米|(H^2+H+1)$ 和 2 的顺序 ${(\mathbb{Z}/米\mathbb{Z})}^{\times }$是奇数，则存在无穷多个素数q使得存在具有上述参数的强正则 Cayley 图。