On ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-$(1+u)$-additive constacyclic

Abstract

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.