Systole Length in Hyperbolic $n$-Manifolds

We show that the length $R$ of a systole of a closed hyperbolic $n$-manifold $(n\ge 3)$ admitting a triangulation by $t$ $n$-simplices can be bounded below by a function of $n$ and $t$, namely $$R\ge \frac{1}{2^{(nt{)}^{O(n^{4}t)}}}\mathrm{.}$$ We do this by finding a relation between the number of $n$-simplices and the diameter of the manifold and by giving explicit bounds for a well known relation between the length of the core curve of a Margulis tube and its radius. We prove the same result for finite volume manifolds, with a similar but slightly more involved proof.