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.

No reviews yet.

No comments yet.