On Codes over Eisenstein Integers

TL;DR

This paper introduces new coding constructions over quotient rings of Eisenstein integers, utilizing set partitioning to improve distance properties and enable multilevel coding for more efficient signal constellations.

Contribution

It presents a novel approach to partition quotient rings of Eisenstein integers into subgroups with enhanced minimum distances, extending coding theory beyond prime-sized fields.

Findings

Partitioning quotient rings improves minimum distances.

Enhanced multilevel coding schemes are feasible.

Signal constellation efficiency is increased.

Abstract

We propose constructions of codes over quotient rings of Eisenstein integers equipped with the Euclidean, square Euclidean, and hexagonal distances as a generalization of codes over Eisenstein integer fields. By set partitioning, we effectively divide the ring of Eisenstein integers into equal-sized subsets for distinct encoding. Unlike in Eisenstein integer fields of prime size, where partitioning is not feasible due to structural limitations, we partition the quotient rings into additive subgroups in such a way that the minimum square Euclidean and hexagonal distances of each subgroup are strictly larger than in the original set. This technique facilitates multilevel coding and enhances signal constellation efficiency.