  There are few explicit examples of two player nonlocal games with a large gap
between classical and quantum value. One of the reasons is that estimating the
classical value is usually a hard computational task. This paper is devoted to
analyzing classical values of the so-called linear games (generalization of XOR
games to a larger number of outputs). We employ nontrivial results from graph
theory and combine them with number theoretic results used earlier in the
context of harmonic analysis to obtain a novel tool -- {\it the girth method}
-- allowing to provide explicit examples of linear games with prescribed low
classical value. In particular, we provide games with minimal possible
classical value. We then speculate on the potential unbounded violation, by
comparing the obtained classical values with a known upper bound for the
quantum value. If this bound can be even asymptotically saturated, our games
would have the best ratio of quantum to classical value as a function of the
product of the number of inputs and outputs when compared to other explicit
(i.e. non-random) constructions.
