On Constacyclic Codes Over $\mathbb{Z}_4[v]/\langle v^2-v \rangle$ and Their Gray Images

In this letter, we discuss the construction of all constacyclic codes over
$R = \mathbb {Z}_{4}[v]/\langle v^{2}-v \rangle $
. Some significant properties of linear codes over
$R$
have been explored. The self-dual constacyclic codes for odd length over
$R$
are determined. Several examples of
$(1 + {2}v)$
- constacyclic codes and
$(3 + {2}v)$
-constacyclic codes over
$R$
, whose
$\mathbb {Z}_{4}$
-images are new
$\mathbb {Z}_{4}$
-linear codes with better parameters, are provided.