Let R be the Galois ring of characteristic 4 and cardinality 4m, where m is a natural number. Let C be a linear code of length n over R andbe the Homogeneous Gray map on Rn. In this paper, we show that (C) is linear if and only if for every X, Y ∈ C, 2(X Y) ∈ C. Using the generator polynomial of a cyclic code of odd length over R, a necessary and sufficient condition is given which its Gray image is linear.
