有限群上函数的自同态和仿射映射的最坏情况逼近
Worst-case approximability of functions on finite groups by endomorphisms and affine maps
摘要：
研究了有限群$G$上一般函数到$G$的群自同态和$G$上仿射映射的集合$\operatorname{End}(G)$和$\operatorname{Aff}(G)$的最大汉明距离（或者说是“最小可逼近性”的补充概念），后者是自同态的某种推广.我们给出了这两个数量的一般界限，并讨论了无限类极值的例子（其中两个汉明距离可以尽可能大）。最后，我们用$|G|\leq15$计算了所有有限群$G$的这两个量的精确值。
英文摘要：
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 $|G|\leq15$.
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