Composition of Gray Isometries

In classical coding theory, Gray isometries are usually defined as mappings between finite Frobenius rings, which include the ring $Z_m$ of integers modulo $m$, and the finite fields. In this paper, we derive an isometric mapping from $Z_8$ to $Z_4^2$ from the composition of the Gray isometries on $Z_8$ and on $Z_4^2$. The image under this composition of a $Z_8$-linear block code of length $n$ with homogeneous distance $d$ is a (not necessarily linear) quaternary block code of length $2n$ with Lee distance $d$.