# On Codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$

## TL;DR

This paper explores the properties and constructions of linear LCD and formally self-dual codes over a specific ring extension of finite fields, providing existence conditions, bounds, and new code constructions.

## Contribution

It introduces new conditions for the existence of LCD codes and constructs formally self-dual codes over the ring R, extending bounds on minimum distances.

## Key findings

- Conditions for the existence of LCD codes over R
- Construction methods for formally self-dual codes over R
- Bounds on minimum distances of LCD codes over finite fields and rings

## Abstract

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $\F_q$ and extend these to codes over $R$.