Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quaternions over Galois rings and their codes   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.

18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an