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

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.