Triprojective almost perfect nonlinear permutations and functions

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 $\mathrm{G}\mathrm{L}(3,2^m)$.