In this paper, we study $\mathbb{ Z}_{ 2}\mathbb{ Z}_{ 2}[u]$-$(1+u)$-additive constacyclic code of arbitrary length. Firstly, we study the algebraic structure of this family of codes and a set of generator polynomials for this family as a $(\mathbb{ Z}_{ 2}+u\mathbb{ Z}_{ 2})[x]$-submodule of the ring $R_{ \alpha,\beta}$. Secondly, we give the minimal generating sets of this family codes, and we determine the relationship of generators between the $\mathbb{ Z}_{ 2}\mathbb{ Z}_{ 2}[u]$-$(1+u)$-additive constacyclic codes and its dual and give the parameters in terms of the degrees of the generator polynomials of the code. Lastly, we also study $\mathbb{ Z}_{ 2}\mathbb{ Z}_{ 2}[u]$-$(1+u)$-additive constacyclic code in terms of the Gray images.
Categories: cs.IT math.IT