On quasi-twisted codes and generalized quasi-twisted codes over $$\mathbb {Z}_{4} +u\mathbb {Z}_{4}$$

In this paper, our main objective is to examine the properties and characteristics of 1-generator \((2 + u)\)-quasi-twisted (QT) codes and \((2 + u)\)-generalized quasi-twisted (GQT) codes over the ring \(\mathbb {Z}_4 +u\mathbb {Z}_4 \), with \(u^2=1\). We determine the structure of the generators and minimal generating sets for both 1-generator \((2 + u)\)-QT and \((2 + u)\)-GQT codes. Additionally, we establish a lower bound for the minimum distance of free 1-generator \((2 + u)\)-QT and \((2 + u)\)-GQT codes over R. Furthermore, we present some numerical examples that illustrate the construction of some optimal \(\mathbb {Z}_4\)-linear codes using the Gray map.