  We construct an explicit family of 3XOR instances which is hard for
$O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to
earlier constructions, which involve a random component, our systems can be
constructed explicitly in deterministic polynomial time.
  Our construction is based on the high-dimensional expanders devised by
Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes,
and our analysis is based on two notions of expansion for these complexes:
cosystolic expansion, and a local isoperimetric inequality due to Gromov.
  Our construction offers an interesting contrast to the recent work of Alev,
Jeronimo and the last author~(FOCS 2019). They showed that 3XOR instances in
which the variables correspond to vertices in a high-dimensional expander are
easy to solve. In contrast, in our instances the variables correspond to the
edges of the complex.
