Parameters of a class of functions over a finite field

We study the class of functions defined on a finite field $GF(q)$ and constructed by means of linear recurrent sequences over the Galois ring $GR(q^n, p^n)$. For this class we investigate: the distances between functions, the distance to the class of affine functions, the number of constructed functions and the number of preimages of elements under action of functions. It is shown that the functions are significantly distant from the class of all affine functions.