  In this paper, we present a new construction of simplicial complexes of
subpolynomial degree with arbitrarily good local spectral expansion.
Previously, the only known high-dimensional expanders (HDXs) with arbitrarily
good expansion and less than polynomial degree were based on one of two
constructions, namely Ramanujan complexes and coset complexes. In contrast, our
construction is a Cayley complex over the group $\mathbb{F}_2^k$, with Cayley
generating set given by a Grassmannian HDX.
  Our construction is in part motivated by a coding-theoretic interpretation of
Grassmannian HDXs that we present, which provides a formal connection between
Grassmannian HDXs, simplicial HDXs, and LDPC codes. We apply this
interpretation to prove a general characterization of the 1-homology groups
over $\mathbb{F}_2$ of Cayley simplicial complexes over $\mathbb{F}_2^k$. Using
this result, we construct simplicial complexes on $N$ vertices with arbitrarily
good local expansion for which the dimension of the 1-homology group grows as
$\Omega(\log^2N)$. No prior constructions in the literature have been shown to
achieve as large a 1-homology group.
