Uniform hypergraphs of girth $6$ and $8$ from generalized polygons

Let $ex_r(N,g)$ be the maximum number of edges in an $r$-uni\-form hypergraph on $N$ vertices with girth at least $g$. We are interested in the asymptotic behavior of this value when $N$ is increasing but parameters $g\in\{6,8\}$ and $r\geq3$ are fixed. It is shown that for some positive constants $c$ and $d$, any integer $r\geq3$ and all sufficiently large integers $N$ the inequalities $ex_r(N,6)\geq N^{\frac{11}{8}-\frac{c}{\sqrt{\log N}}}$ and $ex_r(N,8)\geq N^{\frac{11}{9}-\frac{d}{\sqrt{\log N}}}$ hold.