On Modular Gray Map

This paper introduces an isometry between the modular rings ${\mathbb{Z}}_{2^s}$ and ${\mathbb{Z}}_{2^{s-1}}$ with respect to the homogeneous weights. Certain product of these maps gives Carlet’s generalised Gray map and also Vega’s Gray map. For $s=2$ this reduces to popular Gray map. Several interesting properties of these maps are studied. Towards the end we list several interesting problems to work on.