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Worst-case approximability of functions on finite groups by endomorphisms and affine maps. (English) Zbl 1508.20029

Summary: 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 \(|G|\leq 15\).

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F69 Asymptotic properties of groups

Software:

GAP; SmallGrp
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References:

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