  In this paper we prove that, for any arithmetic hyperbolic $n$-manifold $M$
of the first type, the systole of most of the principal congruence coverings
$M_{I}$ satisfy $$sys_{1}(M_{I})\geq \frac{8}{n(n+1)}\log(vol(M_{I}))-c,$$
where $c$ is a constant independent of $I$. This generalizes previous work of
Buser and Sarnak, and Katz, Schaps and Vishne in dimension 2 and 3. As
applications, we obtain explicit estimates for systolic genus of hyperbolic
manifolds studied by Belolipetsky and the distance of homological codes
constructed by Guth and Lubotzky. In an appendix together with Cayo D\'oria we
prove that the constant $\frac{8}{n(n+1)}$ is sharp.
