  A $\mathrm{CHSH}_{q}$ game is a generalization of the standard two player
$\mathrm{CHSH}$ game, having $q$ different input and output options. In
contrast to the binary game, the best classical and quantum winning strategies
are not known exactly. In this paper we provide a constructive classical
strategy for winning a $\mathrm{CHSH}_{q}$ game, with $q$ being a prime. Our
construction achieves a winning probability better than
$\frac{1}{22}q^{-\frac{2}{3}}$, which is in contrast with the previously known
constructive strategies achieving only the winning probability of $O(q^{-1})$.
