  We obtain algorithmically effective versions of the dense lattice sphere
packings constructed from orders in $\mathbb{Q}$-division rings by the first
author. The lattices in question are lifts of suitable codes from prime
characteristic to orders $\mathcal{O}$ in $\mathbb{Q}$-division rings and we
prove a Minkowski--Hlawka type result for such lifts. Exploiting the additional
symmetries under finite subgroups of units in $\mathcal{O}$, we show this leads
to effective constructions of lattices approaching the best known lower bounds
on the packing density $\Delta_n$ in a variety of new dimensions $n$. This
unifies and extends a number of previous constructions.
