We present a lower bound on the image size of a $d$-uniform map, $d\geq 1$, of finite fields, by extending the methods used for planar maps. In the particularly interesting case of APN maps on binary fields, our bound coincides with the one obtained by Ingo Czerwinski, using a linear programming method. We study properties of APN maps of $\mathbb{F}_{2^n}$ with minimal image set. In particular, we observe that for even $n$, a Dembowski-Ostrom polynomial of form $f(x) =f'(x^3)$ is APN if and only if $f$ is almost-3-to-1, that is when its image set is minimal. We show that any almost-3-to-1 quadratic map is APN, if $n$ is even. For $n$ odd, we present APN Dembowski-Ostrom polynomials on $\mathbb{F}_{2^n}$ with image sizes $ 2^{n-1}$ and $5\cdot 2^{n-3}$. We present several results connecting the image sets of special APN maps with their Walsh spectrum. Especially, we show that a large class of APN maps has the classical Walsh spectrum. Finally, we prove that the image size of a non-bijective almost bent map contains at most $2^n-2^{(n-1)/2}$ elements.

翻译：我们使用线性编程方法,对一张美元统一地图的图像大小显示一个较低的约束值,即$d\geq 1美元,通过扩展平面地图使用的方法,对一个限定字段的图像大小显示一个较低的约束值。对于在二进制字段上的APN地图来说,我们的约束值与Ingo Czerwinski使用线性编程方法获得的匹配值相吻合。我们用最小的图像集来研究 $\ mathbb{F ⁇ 2 ⁇ n}的 APN 地图的属性。我们特别观察到,即使美元(x) =f(x) =f(x) 3) 格式的Dembowski-Ostrom 多元数字($) $(dembowk-Ostrom) 组合值为 $(x) =f(x) =f(x) 3) 美元是APN$(如果美元几乎是三到一美元,也就是其图像集成型的美元,那么,我们的绑定值就与 3- 1级图中的5\cloveal masional masional asion asional $。