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 \(d\).