Quaternions over Galois rings and their codes

TL;DR

This paper explores quaternion rings over Galois rings and Frobenius rings, establishing their properties, weights, and constructing new quaternion codes with optimal distance characteristics.

Contribution

It demonstrates that quaternion rings over Frobenius rings are Frobenius, introduces a homogeneous weight depending on field size, and constructs maximum distance separable quaternion codes.

Findings

Quaternion rings over Frobenius rings are Frobenius.

A homogeneous weight depending on field size is derived.

Constructed quaternion codes that meet the Singleton bound.

Abstract

It is shown in this paper that, if $R$ is a Frobenius ring, then the quaternion ring $\mathcal{H}_{a,b}(R)$ is a Frobenius ring for all units $a,b \in R$. In particular, if $q$ is an odd prime power then $\mathcal{H}_{a,b}(\mathbb{F}_q)$ is the semisimple non-commutative matrix ring $M_2(\mathbb{F}_q)$. Consequently, a homogeneous weight that depends on the field size $q$ is obtained. On the other hand, the homogeneous weight of a finite Frobenius ring with a unique minimal ideal is derived in terms of the size of the ideal. This is illustrated by the quaternions over the Galois ring $GR(2^r,m)$. Finally, one-sided linear block codes over the quaternions over Galois rings are constructed, and certain bounds on the homogeneous distance of the images of these codes are proved. These bounds are based on the Hamming distance of the quaternion code and the parameters of the Galois ring. Good examples of one-sided rate-2/6, 3-quasi-cyclic quaternion codes and their images are generated. One of these codes meets the Singleton bound and is therefore a maximum distance separable code.