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