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

TL;DR

This paper investigates how closely functions on finite groups can be approximated by endomorphisms and affine maps, providing bounds, extremal examples, and exact values for small groups.

Contribution

It introduces bounds on the approximation of functions by endomorphisms and affine maps, and computes exact values for all groups of size up to 15.

Findings

Bounds on minimum Hamming distance to endomorphisms and affine maps

Construction of extremal examples achieving maximum distances

Exact calculations for groups with up to 15 elements

Abstract

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 $\operatorname{End}(G)$ and $\operatorname{Aff}(G)$, 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 $|G|\leq15$.