  In this article, we will show the existence of lattice packings in a sparse
family of dimensions. This construction will be a generalisation of Venkatesh's
lattice packing result. In our construction, we replace the appearance of the
cyclotomic number field with a division algebra over the rational field. For
this, we develop an analogue of Siegel's mean value theorem over lattices that
have a prescribed set of symmetries given by a finite non-commutative group
inside the multiplicative subgroup of a division algebra. This approach
improves the best known lattice packing bounds in many dimensions.
