The Dual and the Gray Image of Codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$

A. Melakheso; K. Guenda

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arXiv · 1504.08097

The Dual and the Gray Image of Codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$

A. Melakheso·K. Guenda

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Abstract

In this paper, we study the linear codes over the commutative ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$ and their Gray images, where $v^{3}=v$. We define the Lee weight of the elements of $R$, we give a Gray map from $R^{n}$ to $\F^{3n}_{q}$ and we give the relation between the dual and the Gray image of a code. This allows us to investigate the structure and properties of self-dual cyclic, formally self-dual and the Gray image of formally self-dual codes over $R$. Further, we give several constructions of formally self-dual codes over $R

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A. Melakheso, K. Guenda. 2015-04-30. The Dual and the Gray Image of Codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$. https://arxiv.org/abs/1504.08097

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