Models & Optimisation and Mathematical Analysis Journal
Volume 10, Numéro 1, Pages 26-29
2022-12-31
$\mathbb{z}_{4}\mathbb{z}_{4}[u^{3}=1]-$cyclic Codes And Their Reversible Codes
Authors : Hebbache Zineb . Sharma Amit .
Abstract
Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides $\mathbb{Z}_{2}\mathbb{Z}_{4}-$additive codes, $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]-$linear codes are introduced as a new member of such families. In this paper, we are interested in a new family of such mixed alphabet codes, i.e., codes over $\mathbb{Z}_{4}\mathbb{Z}_{4}[u]$ with $u^{3}=1,$ we study the structure of cyclic codes over the ring $\mathbb{Z}_{4}R=\mathbb{Z}_{4}(\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4})$ with $u^{3}=1.$ The reversible cyclic codes of arbitrary length over $\mathbb{Z}_{4}R$ are discussed. It is worth noting that the $\mathbb{Z}_{4}-$Gray images are $\mathbb{Z}_{4}-$linear codes. For an application, various examples of reversible cyclic codes over $\mathbb{Z}_{4}R$ are provided, whose $\mathbb{Z}_{4}-$Gray images are $\mathbb{Z}_{4}-$linear codes with good parameters.
Keywords
reversible codes
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