Functions on distance one from APN functions in small number of variables

In this paper, we deal with vectorial Boolean functions $F\colon\mathbb F_2^n\to\mathbb F_2^n$ of dimension $n\geq1$. Functions $F$ and $G$ are EA-nonequivalent if $G\neq A_1\circ F\circ A_2\oplus A$ for any affine functions $A_1$, $A_2$ and $A$, where $A_1$ and $A_2$ are permutations. A function $F$ is called APN if for any $a,b\in\mathbb F_2^n$, where $a$ is nonzero, the equation $F(x)\oplus F(x\oplus a)=b$ has at most two solutions. We prove that there are no APN functions on the distance one from an APN functions up to dimension $5$, from all quadratic APN functions of dimension $6$, and from all known EA-nonequivalent APN functions of dimensions $7$ and $8$.