New upper bounds on the order of mixed cages of girth 6

A $[z,r;g]$-mixed cage is a mixed graph of minimum order such that each vertex has $z$ in-arcs, $z$ out-arcs, $r$ edges, and it has girth $g$, and the minimum order for $[z,r;g]$-mixed graphs is denoted by $n[z,r;g]$. In this paper, we present an infinite family of mixed graphs with girth $6$, that improves, in some cases, the families that we give in G. Araujo-Pardo and L. Mendoza-Cadena. $\mathit{\text{On}}\mathit{\text{Mixed}}\mathit{\text{Cages}}\mathit{\text{of}}\mathit{\text{girth}}\mathit{\text{6}}$, arXiv:2401.14768v2. In particular, if $q$ is an even prime power we construct a family of graphs that satisfies $n[\frac{q}{4},q;6]\le 4q^2-4$, and if $q$ is an odd prime power, and $\frac{q-3}{2}$ is odd then our family satisfies that $n[\frac{q-1}{4},q;6]\le 4q^2-4$, otherwise $n[\frac{q-3}{4},q;6]\le 4q^2-4$.