$({\theta }_i,\lambda )$-constacyclic codes and DNA codes over ${\mathbb{Z}}_4+u{\mathbb{Z}}_4+u^2{\mathbb{Z}}_4$

In this paper, three new automorphisms were identified over the ring ${\mathbb{Z}}_4+u{\mathbb{Z}}_4+u^2{\mathbb{Z}}_4$ where $u^3=u^2$. With the help of these automorphisms, the characteristic structures of the generator polynomials for the ${\theta }_i$-cyclic codes and $({\theta }_i,\lambda )$-constacyclic codes of odd length on this ring were investigated. Also, for all the units over the ring, ${\mathbb{Z}}_4$-images of ${\theta }_i$-cyclic and $({\theta }_i,\lambda )$-constacyclic codes were reviewed with the associated codes based on determined transformations. Using these observations, new and optimal codes were obtained and presented in the table. In addition, a new transformation was identified that involved DNA base pairs with the elements of ${\mathbb{Z}}_4$. Moreover, a unit reverse polynomial was created, and in this way a new generation method has been built to construct reversible DNA codes over this ring. Finally, this article was further enhanced with supporting examples of the DNA as a part of the study.