Explicit affine formulas for distances between tuples in classical discrete structures

Answering a question of Ben Yaacov, Ibarlucía, and Tsankov [5], we show an explicit way to construct an affine formula for the distance between two $n$-tuples in a $\{0,1\}$-valued $\mathrm{\varnothing }$-structure, using $\lceil {\log }_{2}n\rceil$ quantifier alternations.