It is well known that there exist LDPC good error correcting codes (this was proved by Gallager using random methods while explicit constructions were given by Sipser and Spielman ).
The analogous problem for quantum error correcting codes, in spite being an elementary $Z/2Z$-linear algebra problem, is still open.
Simplical complexes and their homology/cohomology give rise to LDPC quantum error correcting codes ( QECC). But all known examples fail to be good.
In a joint work with Larry Guth ( J. Math. Phys. 55, 082202 (2014) ), we constructed a family of LDPC QECC out of congruence quotients of the $4$-dimensional hyperbolic space. Using methods of systolic geometry over $Z/2Z$, we evaluate the parameters of these codes and disprove a conjecture of Ze’mor who predicted that such homological QECC do not exist. The existence of LDPC good QECC is still open.
All notions will be defined and explained.