Article in volume
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Title:
Self-dual cyclic codes over $M_2(\mathbb{Z}_4)$
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 42(2) (2022) 349-362
Received: 2020-10-31 , Revised: 2021-03-24 , Accepted: 2022-05-25 , Available online: 2022-10-05 , https://doi.org/10.7151/dmgaa.1395
Abstract:
In this paper, we study the structure of cyclic codes over$M_2(\mathbb{Z}_4)$
(the matrix ring of matrices of order 2 over $\mathbb{Z}_4$), which is perhaps
the first time that the ring is considered as a code alphabet. This ring is isomorphic
to $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $w$ is a root of the irreducible
polynomial $x^2+x+1 \in \mathbb{Z}_2[x]$ and
$U \cong \left(\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}\right)$.
In our work, we first discuss the structure of the ring $M_2(\mathbb{Z}_4)$
and then focus on the structure of cyclic codes and self-dual cyclic codes over
$M_2(\mathbb{Z}_4)$. Thereafter, we obtain the generators of the cyclic codes
and their dual codes. A few non-trivial examples are given at the end of the
paper.
Keywords:
codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, Gray map, Lee weight, self-dual codes
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