  The linear saturation number $sat^{lin}_k(n,\mathcal{F})$ (linear extremal
number $ex^{lin}_k(n,\mathcal{F})$) of $\mathcal{F}$ is the minimum (maximum)
number of hyperedges of an $n$-vertex linear $k$-uniform hypergraph containing
no member of $\mathcal{F}$ as a subgraph, but the addition of any new hyperedge
such that the result hypergraph is still a linear $k$-uniform hypergraph
creates a copy of some hypergraph in $\mathcal{F}$. Determining $ex_3^{lin}(n$,
Berge-$C_3$) is equivalent to the famous (6,3)-problem, which has been settled
in 1976. Since then, determining the linear extremal numbers of Berge cycles
was extensively studied. As the counterpart of this problem in saturation
problems, the problem of determining the linear saturation numbers of Berge
cycles is considered. In this paper, we prove that $sat^{lin}_k$($n$,
Berge-$C_t)\ge \big\lfloor\frac{n-1}{k-1}\big\rfloor$ for any integers $k\ge3$,
$t\ge 3$, and the equality holds if $t=3$. In addition, we provide an upper
bound for $sat^{lin}_3(n,$ Berge-$C_4)$ and for any disconnected
Berge-$C_4$-saturated linear 3-uniform hypergraph, we give a lower bound for
the number of hyperedges of it.
