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

Let r, s, t be three positive integers and C be a binary linear code of lenght r + s + t. We say that C is a triple cyclic code of lenght (r, s, t) over 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 Z 2 [x]-submodules of Z2[x] x r −1 × Z2[x] x s −1 × Z2[x] x t −1. 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.