Some results about double cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^2\mathbb{F}_{q}$

TL;DR

This paper studies the structure and properties of double cyclic codes over a specific finite ring, providing generating polynomials, matrices, and dual relationships to enhance understanding of their algebraic features.

Contribution

It introduces the polynomial representation, generating matrices, and dual relationships of double cyclic codes over the ring _q+v_q+v^2_q with v^3=v, which is a novel algebraic analysis.

Findings

Explicit generating polynomials for double cyclic codes.

Constructed generating matrices and analyzed their properties.

Explored the relationship between generators and dual codes.

Abstract

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements. This paper mainly researches the polynomial representation of double cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^2\mathbb{F}_{q}$ with $v^3=v$. Firstly, we give the generating polynomials of these double cyclic codes. Secondly, we show the generating matrices of them. Meanwhile, we get quantitative information related to them by the matrix forms. Finally, we investigate the relationship between the generators of double cyclic codes and their duals.