We give a large family of almost perfect nonlinear (APN) permutations of finite vector spaces of every odd dimension divisible by three. We also give APN functions that are not bijective on even dimensions and related highly nonlinear functions. The functions we provide admit a so-called triprojective structure induced by the general linear group $\mathrm{GL}(3,2^m)$.

翻译：我们给出一个大型的几乎完美非线性（APN）置换族，其定义在维数可被3整除的任意奇数维有限向量空间上。我们还给出了在偶数维上非双射的APN函数及其相关的高非线性函数。我们所提供的函数允许由一般线性群$\mathrm{GL}(3,2^m)$诱导的所谓三投影结构。