On quantum codes from cyclic codes over a class of nonchain rings
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In this paper, we extend the results given in \cite{2.5} to a nonchain ring $R_p={\mathbb{F}_p} + v{\mathbb{F}_p} + \cdots + {v^{p - 1}}{\mathbb{F}_p}$, where ${v^p} = v$ and $p$ is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring $R_p$ and study the structure of their duals. We classify cyclic codes containing their duals over $R_p$ by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map $\pi$ defined in \cite{2.7}, we give the parameters of the quantum codes of length $pn$ over $\mathbb{F}_p$ which are obtained from cyclic codes over $R_p.$ Finally, we illustrate the results by giving some examples.
Abstract
In this paper, we extend the results given in \cite{2.5} to a nonchain ring $R_p={\mathbb{F}_p} + v{\mathbb{F}_p} + \cdots + {v^{p - 1}}{\mathbb{F}_p}$, where ${v^p} = v$ and $p$ is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring $R_p$ and study the structure of their duals. We classify cyclic codes containing their duals over $R_p$ by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map $\pi$ defined in \cite{2.7}, we give the parameters of the quantum codes of length $pn$ over $\mathbb{F}_p$ which are obtained from cyclic codes over $R_p.$ Finally, we illustrate the results by giving some examples.
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