Dense Packings via Lifts of Codes to Division Rings

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 that 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.