The Kodaira dimension of Siegel modular varieties of genus 3 or higher

We consider the moduli space $A_{pol}(n)$ of (non-principally) polarised abelian varieties of genus $g\ge 3$ with coprime polarisation and full level-$n$ structure. Based upon the analysis of the Tits building in math/0405321, we give an explicit lower bound on $n$ that is sufficient for the compactified moduli space to be of general type if one further explicit condition is satisfied.