Abstract
We show that if a graph has minimum vertex degree at least d and girth at least g, where (d, g) is (3, 6) or (4, 4), then the incidence system of the graph has a (possibly infinite-dimensional) quantum solution over Z_p for every choice of vertex weights and integer p ≥ 2. In particular, there are linear systems over Z_p, for p an odd prime, such that the corresponding linear system nonlocal game has a perfect commuting-operator strategy, but no perfect classical strategy.