On Modular Gray Map

TL;DR

This paper presents a new isometry between modular rings with respect to homogeneous weights, unifies several Gray maps, and explores their properties and potential research directions.

Contribution

It introduces a novel isometry linking modular rings and generalizes known Gray maps, providing a new perspective and tools for coding theory.

Findings

Established an isometry between Z_{2^s} and Z_{2^{s-1}} with respect to homogeneous weights

Unified Carlet's and Vega's Gray maps through product maps of the introduced isometry

Identified properties of these Gray maps and proposed open problems for future research.

Abstract

This paper introduces an isometry between the modular rings $\Z_{2^s}$ and $\Z_{2^{s-1}}$ 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.