Good Distance Lattices from High Dimensional Expanders

Abstract: 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.