Constacyclic and quasi-twisted codes over $ \mathbb{Z}_{q}[u]/\langle u^{2}-1\rangle $ and new $ \mathbb{Z}_4 $-linear codes

In this paper, we investigate the algebraic structures and properties of constacyclic and quasi-twisted (QT) codes over the ring $ R = \mathbb{Z}_{q}+u\mathbb{Z}_{q} $ with $ u^{2} = 1 $. We show that the image of a constacyclic code over $ R $ under a natural Gray map is a QT code of index $ 2 $ over $ \mathbb{Z}_q $. Given the decomposition of a QT code, we find the decomposition of its dual code. We present 116 new linear codes over $ \mathbb{Z}_{4} $ from the Gray images of QT codes over this ring with $ q = 4 $. Finally, a characterization of linear complementary pair (LCP) constacyclic codes over $ R $ is provided.