Constructive nonlocal games with very small classical values

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.