Values of linear recurring sequences of vectors over finite fields

(1) s(k) = { vk if k ≤ n , ∑n−1 i=0 ais(k − n+ i) if k > n . We note that the elements of the sequence S can be considered as elements of the field Fqn which is an n-dimensional vector space over Fq. It is easy to see that without loss of generality we can suppose that f(0) 6= 0. Thus, it is possible to define the order of f , denoted by τ , as the least positive integer t for which f(x) divides x − 1. It is known from [4] that the period of the sequence S does not exceed τ . For other details concerning polynomials and linear recurring sequence over Fq, see [4]. In this paper we improve and generalize some results from the papers [1], [2], [6], [7] which are also devoted to studying the distribution of values of linear recurring sequences over finite fields. For some applications of such sequences see [1], [2]. It is easy to check that if α, μ ∈ Fqn then for any fixed basis of Fqn over Fq, the coordinate-vectors {ck} of the powers αμ, k = 1, 2, . . . , satisfy such