Worst-case approximability of functions on finite groups by endomorphisms and affine maps

We study the maximum Hamming distance (or rather, the complementary notion of “minimum approximability”) of a general function on a finite group $G$ to either of the sets $\mathrm{End}(G)$ and $\mathrm{Aff}(G)$, of group endomorphisms of $G$ and affine maps on $G$ respectively, the latter being a certain generalization of endomorphisms. We give general bounds on these two quantities and discuss an infinite class of extremal examples (where each of the two Hamming distances can be made as large as generally possible). Finally, we compute the precise values of the two quantities for all finite groups $G$ with $\mathrm{\mid }G\mathrm{\mid }\le 15$.