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

TL;DR

This paper explores linear codes over a specific ring and their Gray images, establishing relationships between dual codes and Gray images, and constructing self-dual codes with potential applications in coding theory.

Contribution

It introduces a Gray map for codes over the ring R and analyzes the duality and structure of self-dual and formally self-dual codes over R.

Findings

Gray map from R^n to F_q^{3n} established

Relationship between dual codes and Gray images analyzed

Constructed several examples of formally self-dual codes

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