On the number of linear uniform hypergraphs with linear girth constraint

For an integer $r\geqslant 3$, a hypergraph on vertex set $[n]$ is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if every two distinct edges share at most one vertex. Given a family $\mathcal{H}$ of linear $r$-uniform hypergraphs,let $Forb_r^L(n,\mathcal{H})$ be the set of linear $r$-uniform hypergraphs on vertex set $[n]$, which does not contain any member from $\mathcal{H}$ as a subgraph. An $r$-uniform linear cycle of length $\ell$, denoted by $C_\ell^r$, is a linear $r$-uniform hypergraph on $(r-1)\ell$ vertices whose edges can be ordered as $\boldsymbol{e}_1,\ldots,\boldsymbol{e}_\ell$ such that $|\boldsymbol{e}_i\cap \boldsymbol{e}_j|=1$ if $j=i\pm 1$ (indices taken modulo $\ell$) and $|\boldsymbol{e}_i\cap \boldsymbol{e}_j|=0$ otherwise. The linear girth of a linear $r$-uniform hypergraph is the smallest integer $\ell$ such that it contains a $C_\ell^r$. Let $Forb_L(n,r,\ell)=Forb_r^L(n,\mathcal{H})$ when $\mathcal{H}=\{C_i^r:\, 3\leqslant i\leqslant \ell\}$, that is, $Forb_L(n,r,\ell)$ is the set of all linear $r$-uniform hypergraphs on $[n]$ with linear girth greater than $\ell$. For integers $r\geqslant 3$ and $\ell\geqslant 4$, Balogh and Li [On the number of linear hypergraphs of large girth, J. Graph Theory, 93(1) (2020), 113-141] showed that $|Forb_L(n,r,\ell)|= 2^{O(n^{1+1/\lfloor \ell/2\rfloor})}$ based on the graph container method. It is natural to obtain $|Forb_L(n,r,\ell)|\geqslant 2^{c\cdot n^{1+1/\ell}}$ for some constant $c$ by probabilistic deletion method. Combined with the known results that $|Forb_L(n,r,3)|= 2^{o (n^{2})}$ and $|Forb_L(n,3,4)|= 2^{\Theta (n^{3/2})}$, by analyzing the random greedy high linear girth linear $r$-uniform hypergraph process, we show $|Forb_L(n,r,\ell)|\geqslant 2^{n^{1+1/(\ell-1)-O(\log\log n/\log n)}}$ for every pair of fixed integers $r,\ell\geqslant 4$, or $r= 3$ and $\ell\geqslant 5$.