The Homogeneous Gray image of linear codes over the Galois ring GR(4, m)

Let R be the Galois ring of characteristic 4 and cardinality \(4^{m}\), where m is a natural number. Let \( \mathcal {C} \) be a linear code of length n over R and \(\Phi \) be the Homogeneous Gray map on \(R^n\). In this paper, we show that \(\Phi (\mathcal {C})\) is linear if and only if for every \(\varvec{X}, \varvec{Y}\in \mathcal {C} \), \(2(\varvec{X} \odot \varvec{Y})\in \mathcal {C}\). 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.