Triple cyclic codes over ${\mathbb{Z}}_2$

Let $r,s,t$ be three positive integers and $\mathcal{C}$ be a binary linear code of lenght $r+s+t$. We say that $\mathcal{C}$ is a triple cyclic code of lenght $(r,s,t)$ over ${\mathbb{Z}}_2$ if the set of coordinates can be partitioned into three parts that any cyclic shift of the coordinates of the parts leaves invariant the code. These codes can be considered as ${\mathbb{Z}}_2[x]$-submodules of $\frac{$\mathbb{Z}$_2[x]}{\langle x^r-1\rangle}\times\frac{$\mathbb{Z}$_2[x]}{\langle x^s-1\rangle}\times\frac{$\mathbb{Z}$_2[x]}{\langle x^t-1\rangle}$. We give the minimal generating sets of this kind of codes. Also, we determine the relationship between the generators of triple cyclic codes and their duals.