  We study a three-fold variant of the hypergraph product code construction,
differing from the standard homological product of three classical codes. When
instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS
quantum LDPC code which might display a large minimum distance. The simplest
instance of this construction, corresponding to the product of 3 repetition
codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon
code. The general construction was introduced in Denise Maurice's PhD thesis,
but has remained poorly understood so far. The reason is that while hypergraph
product codes can be analyzed with combinatorial tools, the XYZ product codes
also depend crucially on the algebraic properties of the parity-check matrices
of the three classical codes, making their analysis much more involved.
  Our main motivation for studying XYZ product codes is that the natural
representatives of logical operators are two-dimensional objects. This
contrasts with standard hypergraph product codes in 3 dimensions which always
admit one-dimensional logical operators. In particular, specific instances of
XYZ product codes with constant rate might display a minimum distance as large
as $\Theta(N^{2/3})$. While we do not prove this result here, we obtain the
dimension of a large class of XYZ product codes, and when restricting to codes
with dimension 1, we reduce the problem of computing the minimum distance to a
more elementary combinatorial problem involving binary 3-tensors. We also
discuss in detail some families of XYZ product codes that can be embedded in
three dimensions with local interaction. Some of these codes seem to share
properties with Haah's cubic codes and might be interesting candidates for
self-correcting quantum memories with a logarithmic energy barrier.
