The linear recurrent sequences over finite fields of largest possible periods are useful in cryptography and coding theory. These sequences are usually generated by homogeneous linear recurrent relations over a finite field (in engineering terminology, such recurrent relations are often known as linear feedback shift registers). We will consider some recent generalizations to such recurrent relations that generate sequence of vectors (instead of just a single element) over a finite field. We shall discuss some results about enumerating these sequences and their relationship with polynomials and matrices over finite fields.