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 End(G) and 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|≤ 15.

中文翻译：

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通过自同态和仿射映射对有限群上的函数进行最坏情况逼近

我们研究有限群上一般函数的最大汉明距离（或者更确切地说，“最小近似性”的补充概念）G到任一集合 End(G) 和 Aff(G), 的群内同态G和仿射地图G，后者分别是自同态的某种概括。我们给出了这两个量的一般界限，并讨论了无限类的极值示例（其中两个汉明距离中的每一个都可以尽可能大）。最后，我们计算所有有限群的两个量的精确值G和|G|≤ 15.