All almost perfect nonlinear (APN) permutations that we know to date admit a special kind of linear self-equivalence, i.e., there exists a permutation $G$ in their CCZ-equivalence class and two linear permutations $A$ and $B$, such that $G \circ A = B \circ G$. After providing a survey on the known APN functions with a focus on the existence of self-equivalences, we explicitly search for APN permutations in dimension 6, 7, and 8 that admit such a linear self-equivalence. In dimension six, we were able to conduct an exhaustive search and obtain that there is only one such APN permutation up to CCZ-equivalence. In dimensions 7 and 8, we performed an exhaustive search for all but a few classes of linear self-equivalences and we did not find any new APN permutation. As one interesting result in dimension 7, we obtain that all APN permutation polynomials with coefficients in $\mathbb{F}_2$ must be (up to CCZ-equivalence) monomial functions.

中文翻译：

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小维线性自等价APN排列

迄今为止，我们知道的所有几乎完美的非线性 (APN) 排列都承认一种特殊的线性自等价，即在它们的 CCZ 等价类中存在一个排列 $G$ 和两个线性排列 $A$ 和 $B$，使得 $G \circ A = B \circ G$。在对已知的 APN 函数进行调查后，重点关注自等价的存在，我们明确地在第 6、7 和 8 维中搜索允许这种线性自等价的 APN 排列。在维度 6 中，我们能够进行详尽的搜索并获得只有一个这样的 APN 排列达到 CCZ 等价。在维度 7 和 8 中，我们对除少数几类线性自等价外的所有类别进行了详尽的搜索，但没有找到任何新的 APN 排列。作为第 7 维的一个有趣结果，