  We describe a new parameterized family of symmetric error-correcting codes
with low-density parity-check matrices (LDPC).
  Our codes can be described in two seemingly different ways. First, in
relation to Reed-Muller codes: our codes are functions on a subset of
$\mathbb{F}^n$ whose restrictions to a prescribed set of affine lines has low
degree. Alternatively, they are Tanner codes on high dimensional expanders,
where the coordinates of the codeword correspond to triangles of a
$2$-dimensional expander, such that around every edge the local view forms a
Reed-Solomon codeword.
  For some range of parameters our codes are provably locally testable, and
their dimension is some fixed power of the block length. For another range of
parameters our codes have distance and dimension that are both linear in the
block length, but we do not know if they are locally testable. The codes also
have the multiplication property: the coordinate-wise product of two codewords
is a codeword in a related code.
  The definition of the codes relies on the construction of a specific family
of simplicial complexes which is a slight variant on the coset complexes of
Kaufman and Oppenheim. We show a novel way to embed the triangles of these
complexes into $\mathbb{F}^n$, with the property that links of edges embed as
affine lines in $\mathbb{F}^n$.
  We rely on this embedding to lower bound the rate of these codes in a way
that avoids constraint-counting and thereby achieves non-trivial rate even when
the local codes themselves have arbitrarily small rate, and in particular below
$1/2$.
