Permutations induced by digit-permutable transformations of a module over a Galois ring of characteristic 4

New classes of nonlinear permutations on a vector space that can be represented by linear transformations of a module over a Galois ring of characteristic 4 are constructed. The system of coordinate functions of such permutation is an orthogonal system of polynomials over the field. Orthogonal systems of two quadratic functions and diagonal quadratic functions are described.