Exact values of girth for some graphs $D(k,q)$ and upper bounds of the order of cage

Let $q$ be a prime power and $k\in\left\{5,7,9,11\right\}$. In this paper it is shown that the girth of a graph $D\left(k,q\right)$ is equal to $k+5$. As a consequence, explicit examples of graphs which provide the best known upper bounds of the order of $\left(r,g\right)$-cages, $r\geq 5$, $g\in\left\{10,14,16\right\}$, are given.