  We show a new framework for constructing good distance lattices from high
dimensional expanders. For error-correcting codes, which have a similar flavor
as lattices, there is a known framework that yields good codes from expanders.
However, prior to our work, there has been no framework that yields good
distance lattices directly from expanders. Interestingly, we need the notion of
high dimensional expansion (and not only one dimensional expansion) for
obtaining large distance lattices which are dense.
  Our construction is obtained by proving the existence of bounded degree high
dimensional cosystolic expanders over any ring, and in particular over
$\mathbb{Z}$. Previous bounded degree cosystolic expanders were known only over
$\mathbb{F}_2$. The proof of the cosystolic expansion over any ring is composed
of two main steps, each of an independent interest: We show that coboundary
expansion over any ring of the links of a bounded degree complex implies that
the complex is a cosystolic expander over any ring. We then prove that all the
links of Ramanujan complexes (which are called spherical buildings) are
coboundary expanders over any ring.
  We follow the strategy of [LMM16] for proving that the spherical building is
a coboundary expander over any ring. Besides of generalizing their proof from
$\mathbb{F}_2$ to any ring, we present it in a detailed way, which might serve
readers with less background who wish to get into the field.

Computational Complexity;Combinatorics