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