Construction of Gray maps for groups of order 16

TL;DR

This paper investigates the construction of Type 1 and Type 2 Gray maps for all groups of order 16, providing explicit constructions and explaining why some groups do not admit Gray maps.

Contribution

It systematically constructs Gray maps for all groups of order 16 and explains the obstructions for certain groups.

Findings

Type 1 Gray maps exist for all groups of order 16

All Type 2 Gray maps for these groups are constructed

Certain groups do not admit Gray maps due to specific reasons

Abstract

In this paper we consider Type 1 Gray maps and Type 2 Gray maps for groups of order 16. First, we confirm that we can construct Type 1 Gray maps for all groups of order 16, and we actually construct them. Next, we confirm that we can construct Type 2 Gray maps for several groups of order 16, and we construct all such maps. Finally we reveal why the other groups do not admit Gray maps.