Bounds for the number of occurrences of elements in a linear recurring sequence over a Galois ring

The number of occurrences of $r$-tuples in the cycle of a linear recurring sequence over a Galois ring is considered. In the special case when the characteristic polynomial of linear recurring sequence is a monic basic irreducible polynomial, we give an upper bound for modulus of difference between the number of occurrences of $r$-tuples in the linear recurring sequence and uniform distributed sequence. In some cases this bound is better than other results which have been obtained for linear recurring sequences of maximal period over residue rings of primary order.