  We propose a family of non-locality unique games for 2 parties based on a
square lattice on an arbitrary surface. We show that, due to structural
similarities with error correction codes of Kitaev for fault tolerant quantum
computation, the games have classical values computable in polynomial time for
$d=2$ measurement outcomes. By representing games in their graph form, for
arbitrary $d$ and underlying surface we provide their classification into
equivalence classes with respect to relabeling of measurement outcomes, for a
selected set of permutations which define the winning conditions. A case study
of games with periodic boundary conditions is presented in order to verify
their impact on classical and quantum values of the family of games. It
suggests that quantum values suffer independently from presence of different
winning conditions that can be imposed due to periodicity, as long as no local
restrictions are in place.

Quantum Physics;Combinatorics